id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-019801 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | State any required conditions first: 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.
Yo... | [
{
"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). (Here the result is $\boxed{\text{Yes}$.) |
math-019802 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Make each step logically reversible (or explain if not): Consider the real matrix
$$A=\begin{pmatrix}-10&1\\0&-11\end{pmatrix}.$$
(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... | [
{
"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... | 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-019803 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Problem: Consider the real matrix
$$A=\begin{pmatrix}-13&1\\0&11\end{pmatrix}.$$
(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 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": "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-019804 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Provide both a computational and a conceptual explanation: Consider the real matrix
$$A=\begin{pmatrix}14&1\\0&6\end{pmatrix}.$$
(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 ... | [
{
"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... | 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-019805 | Linear Algebra: Minimal Polynomial Criterion | 10 | Provide both a computational and a conceptual explanation: Consider the real matrix
$$A=\begin{pmatrix}6&1\\0&-11\end{pmatrix}.$$
(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... | [
{
"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... | 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-019806 | Linear Algebra: Minimal Polynomial Criterion | 10 | Exercise: Consider the real matrix
$$A=\begin{pmatrix}6&1\\0&-20\end{pmatrix}.$$
(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 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... | 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-019807 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Explain each transformation: 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 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... | 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-019808 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Derive the result step-by-step: Consider the real matrix
$$A=\begin{pmatrix}-16&1\\0&-2\end{pmatrix}.$$
(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 ... | [
{
"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-019809 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Answer with a short justification: Consider the real matrix
$$A=\begin{pmatrix}-3&1\\0&18\end{pmatrix}.$$
(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.
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": "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-019810 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Provide a rigorous solution: 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 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": "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). (Here the result is $\boxed{\text{Yes}$.) |
math-019811 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Explain each transformation: Consider the real matrix
$$A=\begin{pmatrix}7&1\\0&10\end{pmatrix}.$$
(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 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). (Here the result is $\boxed{\text{Yes}$.) |
math-019812 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Question: Consider the real matrix
$$A=\begin{pmatrix}7&1\\0&13\end{pmatrix}.$$
(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 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": "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-019813 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Task: Consider the real matrix
$$A=\begin{pmatrix}17&1\\0&17\end{pmatrix}.$$
(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 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": "Cross-check: both derivations land on the same invariant quantity. 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/n... | [
{
"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-019814 | Linear Algebra: Minimal Polynomial Criterion | 10 | Show all reasoning: Consider the real matrix
$$A=\begin{pmatrix}-5&1\\0&6\end{pmatrix}.$$
(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 ... | [
{
"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-019815 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Provide both a computational and a conceptual explanation: Consider the real matrix
$$A=\begin{pmatrix}6&1\\0&17\end{pmatrix}.$$
(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 ... | [
{
"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-019816 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Compute the requested quantity: Consider the real matrix
$$A=\begin{pmatrix}-15&1\\0&10\end{pmatrix}.$$
(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 ... | [
{
"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). (Here the result is $\boxed{\text{Yes}$.) |
math-019817 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Solve and include a self-check: Consider the real matrix
$$A=\begin{pmatrix}-3&1\\0&-8\end{pmatrix}.$$
(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 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": "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-019818 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Be explicit about assumptions: Consider the real matrix
$$A=\begin{pmatrix}17&1\\0&13\end{pmatrix}.$$
(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 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... | 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-019819 | Linear Algebra: Minimal Polynomial Criterion | 10 | Compute the requested quantity: Consider the real matrix
$$A=\begin{pmatrix}9&1\\0&-4\end{pmatrix}.$$
(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 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": "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-019820 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Give a theorem-based solution: Consider the real matrix
$$A=\begin{pmatrix}19&1\\0&13\end{pmatrix}.$$
(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 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). (Here the result is $\boxed{\text{Yes}$.) |
math-019821 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Prompt: Consider the real matrix
$$A=\begin{pmatrix}16&1\\0&-14\end{pmatrix}.$$
(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 justification must expli... | [
{
"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-019822 | Linear Algebra: Minimal Polynomial Criterion | 10 | Work carefully and justify each inference: Consider the real matrix
$$A=\begin{pmatrix}16&1\\0&17\end{pmatrix}.$$
(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 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": "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-019823 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Carefully track domains: 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 justifi... | [
{
"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-019824 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Provide a rigorous solution: 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 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": "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-019825 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Give reasoning, not just computation: 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.
... | [
{
"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-019826 | Linear Algebra: Minimal Polynomial Criterion | 10 | Use two approaches if possible: Consider the real matrix
$$A=\begin{pmatrix}-9&1\\0&-16\end{pmatrix}.$$
(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 ... | [
{
"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-019827 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Solve and sanity-check: Consider the real matrix
$$A=\begin{pmatrix}-19&1\\0&15\end{pmatrix}.$$
(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 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": "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-019828 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Work carefully and justify each inference: Consider the real matrix
$$A=\begin{pmatrix}18&1\\0&-4\end{pmatrix}.$$
(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 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-019829 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Keep the final answer in boxed form: Consider the real matrix
$$A=\begin{pmatrix}11&1\\0&10\end{pmatrix}.$$
(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.
Y... | [
{
"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-019830 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Checkpoint: Consider the real matrix
$$A=\begin{pmatrix}19&1\\0&-13\end{pmatrix}.$$
(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-019831 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Track units/moduli carefully: Consider the real matrix
$$A=\begin{pmatrix}-11&1\\0&4\end{pmatrix}.$$
(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 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-019832 | Linear Algebra: Minimal Polynomial Criterion | 10 | Solve and justify each step: Consider the real matrix
$$A=\begin{pmatrix}-13&1\\0&2\end{pmatrix}.$$
(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 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... | 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-019833 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Do not skip justification steps: Consider the real matrix
$$A=\begin{pmatrix}-5&1\\0&-13\end{pmatrix}.$$
(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": "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-019834 | Linear Algebra: Minimal Polynomial Criterion | 10 | Solve and include a self-check: Consider the real matrix
$$A=\begin{pmatrix}-2&1\\0&-16\end{pmatrix}.$$
(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 ... | [
{
"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-019835 | Linear Algebra: Minimal Polynomial Criterion | 10 | Solve and then verify: 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 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... | 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-019836 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Give a fully justified solution: Consider the real matrix
$$A=\begin{pmatrix}5&1\\0&13\end{pmatrix}.$$
(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 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": "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-019837 | Linear Algebra: Minimal Polynomial Criterion | 10 | Problem: Consider the real matrix
$$A=\begin{pmatrix}-14&1\\0&-17\end{pmatrix}.$$
(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": "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-019838 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Do not skip justification steps: Consider the real matrix
$$A=\begin{pmatrix}-14&1\\0&12\end{pmatrix}.$$
(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": "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-019839 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Solve with verification: Consider the real matrix
$$A=\begin{pmatrix}11&1\\0&5\end{pmatrix}.$$
(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 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": "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). (Here the result is $\boxed{\text{Yes}$.) |
math-019840 | Linear Algebra: Minimal Polynomial Criterion | 10 | Give a theorem-based solution: Consider the real matrix
$$A=\begin{pmatrix}10&1\\0&12\end{pmatrix}.$$
(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 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... | 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-019841 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Explain why your operations are valid: Consider the real matrix
$$A=\begin{pmatrix}10&1\\0&14\end{pmatrix}.$$
(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.
... | [
{
"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-019842 | Linear Algebra: Minimal Polynomial Criterion | 10 | Solve and sanity-check: Consider the real matrix
$$A=\begin{pmatrix}-8&1\\0&3\end{pmatrix}.$$
(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 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... | 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-019843 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Work carefully and justify each inference: 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 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": "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-019844 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Try to avoid pattern-matching; explain why: Consider the real matrix
$$A=\begin{pmatrix}19&1\\0&17\end{pmatrix}.$$
(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 criter... | [
{
"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-019845 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Write the solution set clearly: Consider the real matrix
$$A=\begin{pmatrix}-18&1\\0&16\end{pmatrix}.$$
(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 ... | [
{
"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-019846 | Linear Algebra: Minimal Polynomial Criterion | 10 | Task: Consider the real matrix
$$A=\begin{pmatrix}7&1\\0&-6\end{pmatrix}.$$
(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 explicitl... | [
{
"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-019847 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Answer with a short justification: Consider the real matrix
$$A=\begin{pmatrix}-6&1\\0&16\end{pmatrix}.$$
(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.
You... | [
{
"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-019848 | Linear Algebra: Minimal Polynomial Criterion | 10 | Proceed methodically: Consider the real matrix
$$A=\begin{pmatrix}-11&1\\0&-1\end{pmatrix}.$$
(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 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... | 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-019849 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Explain each transformation: Consider the real matrix
$$A=\begin{pmatrix}-16&1\\0&5\end{pmatrix}.$$
(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 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": "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-019850 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Solve and sanity-check: Consider the real matrix
$$A=\begin{pmatrix}-9&1\\0&17\end{pmatrix}.$$
(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 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... | 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-019851 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Keep the final answer in boxed form: 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.
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": "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-019852 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Question: 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.
Your justification must expl... | [
{
"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-019853 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Problem: Consider the real matrix
$$A=\begin{pmatrix}-4&1\\0&3\end{pmatrix}.$$
(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 explic... | [
{
"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-019854 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Indicate where a theorem is used: Consider the real matrix
$$A=\begin{pmatrix}6&1\\0&20\end{pmatrix}.$$
(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 ... | [
{
"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-019855 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Answer with a short justification: Consider the real matrix
$$A=\begin{pmatrix}9&1\\0&6\end{pmatrix}.$$
(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 ... | [
{
"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). |
math-019856 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Prompt: Consider the real matrix
$$A=\begin{pmatrix}14&1\\0&-9\end{pmatrix}.$$
(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 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": "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-019857 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Track quantifiers carefully: Consider the real matrix
$$A=\begin{pmatrix}-8&1\\0&14\end{pmatrix}.$$
(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 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... | 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-019858 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Keep the final answer in boxed form: 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.
Yo... | [
{
"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). |
math-019859 | Linear Algebra: Minimal Polynomial Criterion | 10 | Proceed methodically: Consider the real matrix
$$A=\begin{pmatrix}0&1\\0&10\end{pmatrix}.$$
(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 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": "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-019860 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Find the exact value: 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.
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... | 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-019861 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Give a theorem-based solution: 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.
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... | 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-019862 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Prompt: Consider the real matrix
$$A=\begin{pmatrix}13&1\\0&7\end{pmatrix}.$$
(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 must explici... | [
{
"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... | 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-019863 | Linear Algebra: Minimal Polynomial Criterion | 10 | Determine the requested value: Consider the real matrix
$$A=\begin{pmatrix}6&1\\0&8\end{pmatrix}.$$
(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 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": "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-019864 | Linear Algebra: Minimal Polynomial Criterion | 10 | Solve and justify each step: Consider the real matrix
$$A=\begin{pmatrix}-6&1\\0&-11\end{pmatrix}.$$
(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 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": "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-019865 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Give reasoning, not just computation: Consider the real matrix
$$A=\begin{pmatrix}-9&1\\0&-4\end{pmatrix}.$$
(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": "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-019866 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Show all reasoning: Consider the real matrix
$$A=\begin{pmatrix}-13&1\\0&-5\end{pmatrix}.$$
(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 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... | 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-019867 | Linear Algebra: Minimal Polynomial Criterion | 10 | Challenge: Consider the real matrix
$$A=\begin{pmatrix}-5&1\\0&-20\end{pmatrix}.$$
(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 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... | 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-019868 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Explain why your operations are valid: Consider the real matrix
$$A=\begin{pmatrix}9&1\\0&18\end{pmatrix}.$$
(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.
... | [
{
"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-019869 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Carefully track domains: Consider the real matrix
$$A=\begin{pmatrix}16&1\\0&11\end{pmatrix}.$$
(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 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": "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-019870 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Write the solution set clearly: Consider the real matrix
$$A=\begin{pmatrix}-4&1\\0&8\end{pmatrix}.$$
(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": "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-019871 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Use two approaches if possible: Consider the real matrix
$$A=\begin{pmatrix}19&1\\0&-15\end{pmatrix}.$$
(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 ... | [
{
"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-019872 | Linear Algebra: Minimal Polynomial Criterion | 10 | Write the solution set clearly: Consider the real matrix
$$A=\begin{pmatrix}9&1\\0&-16\end{pmatrix}.$$
(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 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": "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-019873 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Give a fully justified solution: Consider the real matrix
$$A=\begin{pmatrix}15&1\\0&-15\end{pmatrix}.$$
(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... | [
{
"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-019874 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Try to avoid pattern-matching; explain why: Consider the real matrix
$$A=\begin{pmatrix}15&1\\0&8\end{pmatrix}.$$
(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 criteri... | [
{
"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-019875 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Prompt: Consider the real matrix
$$A=\begin{pmatrix}17&1\\0&-10\end{pmatrix}.$$
(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": "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-019876 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Provide both a computational and a conceptual explanation: Consider the real matrix
$$A=\begin{pmatrix}5&1\\0&16\end{pmatrix}.$$
(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 ... | [
{
"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-019877 | Linear Algebra: Minimal Polynomial Criterion | 10 | Where appropriate, name the theorem you use: Consider the real matrix
$$A=\begin{pmatrix}-13&1\\0&-20\end{pmatrix}.$$
(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 cri... | [
{
"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... | 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-019878 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Track quantifiers carefully: Consider the real matrix
$$A=\begin{pmatrix}8&1\\0&9\end{pmatrix}.$$
(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 justif... | [
{
"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-019879 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Task: Consider the real matrix
$$A=\begin{pmatrix}19&1\\0&5\end{pmatrix}.$$
(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 explicitl... | [
{
"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-019880 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | State any required conditions first: 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.
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": "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-019881 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Question: Consider the real matrix
$$A=\begin{pmatrix}-6&1\\0&4\end{pmatrix}.$$
(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 expli... | [
{
"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... | 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-019882 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Answer using clear logical steps: Consider the real matrix
$$A=\begin{pmatrix}-18&1\\0&7\end{pmatrix}.$$
(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... | [
{
"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-019883 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Solve and include a self-check: Consider the real matrix
$$A=\begin{pmatrix}-1&1\\0&-2\end{pmatrix}.$$
(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": "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-019884 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Give reasoning, not just computation: Consider the real matrix
$$A=\begin{pmatrix}-16&1\\0&-9\end{pmatrix}.$$
(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.
... | [
{
"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-019885 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Give a fully justified solution: Consider the real matrix
$$A=\begin{pmatrix}3&1\\0&-17\end{pmatrix}.$$
(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 ... | [
{
"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). (Here the result is $\boxed{\text{Yes}$.) |
math-019886 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Answer using clear logical 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": "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-019887 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Determine the requested value: Consider the real matrix
$$A=\begin{pmatrix}4&1\\0&-18\end{pmatrix}.$$
(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 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... | 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-019888 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Task: Consider the real matrix
$$A=\begin{pmatrix}-6&1\\0&9\end{pmatrix}.$$
(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 must explicitl... | [
{
"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... | 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-019889 | Linear Algebra: Minimal Polynomial Criterion | 10 | Show all reasoning: Consider the real matrix
$$A=\begin{pmatrix}18&1\\0&-16\end{pmatrix}.$$
(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 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... | 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-019890 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Derive the result step-by-step: Consider the real matrix
$$A=\begin{pmatrix}-4&1\\0&-10\end{pmatrix}.$$
(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 ... | [
{
"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-019891 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Carefully track domains: 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 justifi... | [
{
"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). |
math-019892 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Work this out carefully: Consider the real matrix
$$A=\begin{pmatrix}8&1\\0&-12\end{pmatrix}.$$
(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 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": "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-019893 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Solve and sanity-check: 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 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... | 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-019894 | Linear Algebra: Jordan Form Intuition (2×2) | 10 | Determine the requested value: Consider the real matrix
$$A=\begin{pmatrix}-7&1\\0&1\end{pmatrix}.$$
(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": "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-019895 | Linear Algebra: Minimal Polynomial Criterion | 10 | Give a theorem-based solution: 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 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-019896 | Linear Algebra: Minimal Polynomial Criterion | 10 | Find the exact value: Consider the real matrix
$$A=\begin{pmatrix}-17&1\\0&6\end{pmatrix}.$$
(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 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": "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-019897 | Linear Algebra: Minimal Polynomial Criterion | 10 | Task: Consider the real matrix
$$A=\begin{pmatrix}-7&1\\0&-13\end{pmatrix}.$$
(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 explici... | [
{
"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-019898 | Linear Algebra: Diagonalizability — Eigenvectors | 10 | Explain each transformation: Consider the real matrix
$$A=\begin{pmatrix}2&1\\0&-11\end{pmatrix}.$$
(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 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). |
math-019899 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Problem: Consider the real matrix
$$A=\begin{pmatrix}-3&1\\0&5\end{pmatrix}.$$
(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": "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). (Here the result is $\boxed{\text{Yes}$.) |
math-019900 | Linear Algebra: Algebraic vs Geometric Multiplicity | 10 | Carefully track domains: Consider the real matrix
$$A=\begin{pmatrix}-15&1\\0&6\end{pmatrix}.$$
(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 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": "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 $\boxed{\text{Yes}$.) |
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