id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-019701 | Linear Algebra: Minimal Polynomial Criterion | 10 | Do not skip justification steps: Consider the real matrix
$$A=\begin{pmatrix}-15&1\\0&-9\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019702 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Complete the analysis: Consider the real matrix
$$A=\begin{pmatrix}14&1\\0&-5\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justificat... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019703 | Linear Algebra: Minimal Polynomial Criterion | 10 | Explain why your operations are valid: Consider the real matrix
$$A=\begin{pmatrix}-8&1\\0&19\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019704 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Do not skip justification steps: Consider the real matrix
$$A=\begin{pmatrix}-11&1\\0&12\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019705 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Determine the requested value: Consider the real matrix
$$A=\begin{pmatrix}7&1\\0&20\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your jus... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019706 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Prompt: Consider the real matrix
$$A=\begin{pmatrix}-5&1\\0&-5\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must explic... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustne... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019707 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Give an answer and a quick verification: Consider the real matrix
$$A=\begin{pmatrix}-18&1\\0&10\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterio... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019708 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Do not skip justification steps: Consider the real matrix
$$A=\begin{pmatrix}-16&1\\0&13\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019709 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Problem: Consider the real matrix
$$A=\begin{pmatrix}10&1\\0&17\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must expli... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019710 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Show all reasoning: Consider the real matrix
$$A=\begin{pmatrix}8&1\\0&-7\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification ... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019711 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Be explicit about assumptions: Consider the real matrix
$$A=\begin{pmatrix}-3&1\\0&6\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your jus... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019712 | Linear Algebra: Minimal Polynomial Criterion | 10 | Problem: Consider the real matrix
$$A=\begin{pmatrix}-7&1\\0&11\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must expli... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019713 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Answer with a short justification: Consider the real matrix
$$A=\begin{pmatrix}2&1\\0&-4\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019714 | Linear Algebra: Minimal Polynomial Criterion | 10 | Show all reasoning: Consider the real matrix
$$A=\begin{pmatrix}-16&1\\0&-6\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justificatio... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019715 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Give an answer and a quick verification: Consider the real matrix
$$A=\begin{pmatrix}-12&1\\0&-7\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterio... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019716 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Give a theorem-based solution: Consider the real matrix
$$A=\begin{pmatrix}-17&1\\0&19\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your j... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019717 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Give an answer and a quick verification: Consider the real matrix
$$A=\begin{pmatrix}4&1\\0&12\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019718 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Work carefully and justify each inference: Consider the real matrix
$$A=\begin{pmatrix}14&1\\0&11\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criteri... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019719 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Checkpoint: Consider the real matrix
$$A=\begin{pmatrix}18&1\\0&-13\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must e... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019720 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Be explicit about assumptions: Consider the real matrix
$$A=\begin{pmatrix}3&1\\0&-1\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your jus... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019721 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Proceed methodically: Consider the real matrix
$$A=\begin{pmatrix}-12&1\\0&18\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justificat... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019722 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Give an answer and a quick verification: Consider the real matrix
$$A=\begin{pmatrix}2&1\\0&2\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the s... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019723 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Where appropriate, name the theorem you use: Consider the real matrix
$$A=\begin{pmatrix}7&1\\0&2\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criteri... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019724 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Compute the requested quantity: Consider the real matrix
$$A=\begin{pmatrix}14&1\\0&-2\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your j... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019725 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Challenge: Consider the real matrix
$$A=\begin{pmatrix}-9&1\\0&-3\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must exp... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019726 | Linear Algebra: Minimal Polynomial Criterion | 10 | Work carefully and justify each inference: Consider the real matrix
$$A=\begin{pmatrix}8&1\\0&18\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterio... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019727 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Challenge: Consider the real matrix
$$A=\begin{pmatrix}19&1\\0&12\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must exp... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019728 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Challenge: Consider the real matrix
$$A=\begin{pmatrix}-11&1\\0&2\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must exp... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019729 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Provide a rigorous solution: Consider the real matrix
$$A=\begin{pmatrix}-20&1\\0&-6\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your jus... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019730 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Complete the analysis: Consider the real matrix
$$A=\begin{pmatrix}5&1\\0&19\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justificati... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019731 | Linear Algebra: Minimal Polynomial Criterion | 10 | Explain what is being counted/optimized: Consider the real matrix
$$A=\begin{pmatrix}-17&1\\0&4\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019732 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Give an answer and a quick verification: Consider the real matrix
$$A=\begin{pmatrix}9&1\\0&-3\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019733 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Explain each transformation: Consider the real matrix
$$A=\begin{pmatrix}1&1\\0&20\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justi... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019734 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Work this out carefully: Consider the real matrix
$$A=\begin{pmatrix}11&1\\0&-8\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justific... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019735 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Solve and sanity-check: Consider the real matrix
$$A=\begin{pmatrix}-3&1\\0&12\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justifica... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019736 | Linear Algebra: Minimal Polynomial Criterion | 10 | Explain what is being counted/optimized: Consider the real matrix
$$A=\begin{pmatrix}13&1\\0&16\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019737 | Linear Algebra: Minimal Polynomial Criterion | 10 | Proceed methodically: Consider the real matrix
$$A=\begin{pmatrix}-13&1\\0&-16\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justifica... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019738 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Track units/moduli carefully: Consider the real matrix
$$A=\begin{pmatrix}14&1\\0&14\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your jus... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustne... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{No}$.) |
math-019739 | Linear Algebra: Minimal Polynomial Criterion | 10 | Exercise: Consider the real matrix
$$A=\begin{pmatrix}12&1\\0&-2\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must expl... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019740 | Linear Algebra: Minimal Polynomial Criterion | 10 | Question: Consider the real matrix
$$A=\begin{pmatrix}-3&1\\0&15\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must expl... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019741 | Linear Algebra: Minimal Polynomial Criterion | 10 | Provide a rigorous solution: Consider the real matrix
$$A=\begin{pmatrix}17&1\\0&-9\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your just... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019742 | Linear Algebra: Minimal Polynomial Criterion | 10 | Proceed methodically: Consider the real matrix
$$A=\begin{pmatrix}-9&1\\0&15\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justificati... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019743 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Question: Consider the real matrix
$$A=\begin{pmatrix}-1&1\\0&-19\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must exp... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019744 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Prompt: Consider the real matrix
$$A=\begin{pmatrix}12&1\\0&-10\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must expli... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019745 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Show all reasoning: Consider the real matrix
$$A=\begin{pmatrix}18&1\\0&-17\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justificatio... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019746 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Use two approaches if possible: Consider the real matrix
$$A=\begin{pmatrix}5&1\\0&8\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your jus... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019747 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Show all reasoning: Consider the real matrix
$$A=\begin{pmatrix}10&1\\0&1\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification ... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019748 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Track quantifiers carefully: Consider the real matrix
$$A=\begin{pmatrix}-3&1\\0&1\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justi... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019749 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Solve and sanity-check: Consider the real matrix
$$A=\begin{pmatrix}2&1\\0&-13\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justifica... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019750 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Warm-up: Consider the real matrix
$$A=\begin{pmatrix}-13&1\\0&-17\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must exp... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019751 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Write the solution set clearly: Consider the real matrix
$$A=\begin{pmatrix}13&1\\0&10\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your j... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019752 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Proceed methodically: Consider the real matrix
$$A=\begin{pmatrix}-16&1\\0&-16\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justifica... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"ro... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{No}$.) |
math-019753 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Keep the final answer in boxed form: Consider the real matrix
$$A=\begin{pmatrix}13&1\\0&-3\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Y... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019754 | Linear Algebra: Minimal Polynomial Criterion | 10 | Where appropriate, name the theorem you use: Consider the real matrix
$$A=\begin{pmatrix}-17&1\\0&2\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named crite... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019755 | Linear Algebra: Minimal Polynomial Criterion | 10 | Proceed methodically: Consider the real matrix
$$A=\begin{pmatrix}2&1\\0&5\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019756 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Proceed methodically: Consider the real matrix
$$A=\begin{pmatrix}-10&1\\0&-6\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justificat... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019757 | Linear Algebra: Minimal Polynomial Criterion | 10 | Be explicit about assumptions: Consider the real matrix
$$A=\begin{pmatrix}20&1\\0&-18\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your j... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019758 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Determine the requested value: Consider the real matrix
$$A=\begin{pmatrix}-12&1\\0&-5\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your j... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019759 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Try to avoid pattern-matching; explain why: Consider the real matrix
$$A=\begin{pmatrix}7&1\\0&7\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterio... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustne... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{No}$.) |
math-019760 | Linear Algebra: Minimal Polynomial Criterion | 10 | Try to avoid pattern-matching; explain why: Consider the real matrix
$$A=\begin{pmatrix}0&1\\0&3\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterio... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019761 | Linear Algebra: Minimal Polynomial Criterion | 10 | Proceed methodically: Consider the real matrix
$$A=\begin{pmatrix}1&1\\0&9\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019762 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Use two approaches if possible: Consider the real matrix
$$A=\begin{pmatrix}-4&1\\0&0\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your ju... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019763 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Explain why your operations are valid: Consider the real matrix
$$A=\begin{pmatrix}11&1\\0&-2\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019764 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Give a theorem-based solution: Consider the real matrix
$$A=\begin{pmatrix}1&1\\0&15\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your jus... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019765 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Answer with a short justification: Consider the real matrix
$$A=\begin{pmatrix}20&1\\0&-11\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Yo... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019766 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Give reasoning, not just computation: Consider the real matrix
$$A=\begin{pmatrix}18&1\\0&-3\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019767 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Checkpoint: Consider the real matrix
$$A=\begin{pmatrix}-13&1\\0&4\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must ex... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019768 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Keep the final answer in boxed form: Consider the real matrix
$$A=\begin{pmatrix}-7&1\\0&-16\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019769 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Be explicit about assumptions: Consider the real matrix
$$A=\begin{pmatrix}-6&1\\0&-3\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your ju... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019770 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Solve and justify each step: Consider the real matrix
$$A=\begin{pmatrix}-12&1\\0&13\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your jus... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019771 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Task: Consider the real matrix
$$A=\begin{pmatrix}-16&1\\0&1\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must explicit... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019772 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Make each step logically reversible (or explain if not): Consider the real matrix
$$A=\begin{pmatrix}0&1\\0&0\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a n... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the s... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{No}$.) |
math-019773 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Provide a rigorous solution: Consider the real matrix
$$A=\begin{pmatrix}9&1\\0&11\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justi... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019774 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Show all reasoning: Consider the real matrix
$$A=\begin{pmatrix}-13&1\\0&17\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justificatio... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019775 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Provide a rigorous solution: Consider the real matrix
$$A=\begin{pmatrix}-7&1\\0&15\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your just... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019776 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Write the solution set clearly: Consider the real matrix
$$A=\begin{pmatrix}8&1\\0&-8\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your ju... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019777 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Derive the result step-by-step: Consider the real matrix
$$A=\begin{pmatrix}-3&1\\0&-14\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your ... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019778 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Solve and then verify: Consider the real matrix
$$A=\begin{pmatrix}-4&1\\0&-13\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justifica... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019779 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Carefully track domains: Consider the real matrix
$$A=\begin{pmatrix}5&1\\0&2\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justificat... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019780 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Explain what is being counted/optimized: Consider the real matrix
$$A=\begin{pmatrix}-4&1\\0&16\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019781 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Solve and sanity-check: Consider the real matrix
$$A=\begin{pmatrix}16&1\\0&-1\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justifica... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019782 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Carefully track domains: Consider the real matrix
$$A=\begin{pmatrix}-8&1\\0&20\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justific... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019783 | Linear Algebra: Minimal Polynomial Criterion | 10 | Solve and then verify: Consider the real matrix
$$A=\begin{pmatrix}-9&1\\0&5\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justificati... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019784 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Explain why your operations are valid: Consider the real matrix
$$A=\begin{pmatrix}-16&1\\0&15\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019785 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Answer using clear logical steps: Consider the real matrix
$$A=\begin{pmatrix}20&1\\0&-20\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
You... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019786 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | State any required conditions first: Consider the real matrix
$$A=\begin{pmatrix}-13&1\\0&-15\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019787 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Solve and include a self-check: Consider the real matrix
$$A=\begin{pmatrix}-14&1\\0&-8\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your ... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019788 | Linear Algebra: Minimal Polynomial Criterion | 10 | Solve with verification: Consider the real matrix
$$A=\begin{pmatrix}-10&1\\0&2\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justific... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019789 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Track units/moduli carefully: Consider the real matrix
$$A=\begin{pmatrix}-12&1\\0&-17\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your j... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019790 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Problem: Consider the real matrix
$$A=\begin{pmatrix}-9&1\\0&2\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must explic... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019791 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Prompt: Consider the real matrix
$$A=\begin{pmatrix}3&1\\0&6\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must explicit... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019792 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Solve and then verify: Consider the real matrix
$$A=\begin{pmatrix}-2&1\\0&4\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justificati... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019793 | Linear Algebra: Minimal Polynomial Criterion | 10 | Compute the requested quantity: Consider the real matrix
$$A=\begin{pmatrix}-10&1\\0&-4\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your ... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019794 | Linear Algebra: Minimal Polynomial Criterion | 10 | Task: Consider the real matrix
$$A=\begin{pmatrix}6&1\\0&-15\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must explicit... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019795 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Challenge: Consider the real matrix
$$A=\begin{pmatrix}-2&1\\0&-3\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must exp... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019796 | Linear Algebra: Minimal Polynomial Criterion | 10 | Prompt: Consider the real matrix
$$A=\begin{pmatrix}-20&1\\0&-12\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must expl... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019797 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Answer with a short justification: Consider the real matrix
$$A=\begin{pmatrix}17&1\\0&-4\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
You... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-019798 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Try to avoid pattern-matching; explain why: Consider the real matrix
$$A=\begin{pmatrix}1&1\\0&1\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterio... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the s... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{No}$.) |
math-019799 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Explain what is being counted/optimized: Consider the real matrix
$$A=\begin{pmatrix}-11&1\\0&-7\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterio... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-019800 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Answer using clear logical steps: Consider the real matrix
$$A=\begin{pmatrix}7&1\\0&-8\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your ... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
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