id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-017701 | Linear Algebra: Determinants — Cross-Validation | 9 | Solve and sanity-check: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&4&5\\-5&-3&-4\\4&-1&-6\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-15}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-017702 | Linear Algebra: Minimal Polynomial Criterion | 9 | Solve and justify each step: Consider the real matrix
$$A=\begin{pmatrix}-11&1\\0&9\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your just... | [
{
"method_name": "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-017703 | Linear Algebra: Determinants — Row Operations | 9 | Solve and then verify: Compute the determinant of the matrix
$$A=\begin{pmatrix}0&6&-2\\-3&-2&-5\\2&-6&0\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-104}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number."... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-017704 | Linear Algebra: Minimal Polynomial Criterion | 9 | Exercise: 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 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": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"ro... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-017705 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | Make each step logically reversible (or explain if not): 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 n... | [
{
"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-017706 | Linear Algebra: Determinants — Cross-Validation | 9 | Prompt: Compute the determinant of the matrix
$$A=\begin{pmatrix}0&-1&1\\-4&5&-3\\-1&3&-3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.
When using ro... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{2}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-017707 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Solve and then verify: Consider the real matrix
$$A=\begin{pmatrix}4&1\\0&-11\end{pmatrix}.$$
(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 justificat... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | 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-017708 | Linear Algebra: Determinants — Cross-Validation | 9 | Track units/moduli carefully: Compute the determinant of the matrix
$$A=\begin{pmatrix}-2&5&1\\0&-6&3\\0&6&-4\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must a... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-12}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both metho... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-017709 | Linear Algebra: Determinants — Cross-Validation | 9 | Give reasoning, not just computation: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&2&2\\-1&5&6\\3&4&6\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods ... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{22}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",
... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{22}$.) |
math-017710 | Linear Algebra: Minimal Polynomial Criterion | 9 | Provide a rigorous solution: 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 a named criterion.
Your jus... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "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-017711 | Linear Algebra: Determinants — Row Operations | 9 | Show all reasoning: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&5&3\\4&-6&6\\5&-2&6\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.
When... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{0}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Takeaway: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{0}$.) |
math-017712 | Linear Algebra: Determinants — Cross-Validation | 9 | Work carefully and justify each inference: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&1&1\\-1&-3&2\\2&1&-4\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two m... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{65}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",
... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-017713 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Derive the result step-by-step: Consider the real matrix
$$A=\begin{pmatrix}-14&1\\0&1\end{pmatrix}.$$
(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 j... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-017714 | Matrix Theory: Determinant Properties | 9 | Solve and then verify: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&6&5\\1&-5&-4\\-6&6&4\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.
... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{4}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",
... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{4}$.) |
math-017715 | Matrix Theory: Determinant Properties | 9 | Write the solution set clearly: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&6&5\\2&-1&4\\1&-2&-3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{89}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",
... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{89}$.) |
math-017716 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Solve and then verify: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&-3&-5\\4&-6&-3\\0&-4&-5\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{92}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both method... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{92}$.) |
math-017717 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Derive the result step-by-step: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&-3&-3\\1&3&-6\\2&3&3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{162}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both metho... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{162}$.) |
math-017718 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Compute the requested quantity: 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.
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). |
math-017719 | Linear Algebra: Determinants — Cross-Validation | 9 | Explain what is being counted/optimized: Compute the determinant of the matrix
$$A=\begin{pmatrix}-2&5&-4\\2&-2&5\\-3&3&-2\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two m... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-33}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same num... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-33}$.) |
math-017720 | Linear Algebra: Determinants — Cross-Validation | 9 | Solve and include a self-check: Compute the determinant of the matrix
$$A=\begin{pmatrix}-5&2&2\\5&3&1\\-6&-6&-1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods mus... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-41}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-41}$.) |
math-017721 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Problem: Consider the real matrix
$$A=\begin{pmatrix}-6&1\\0&-6\end{pmatrix}.$$
(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 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": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"ro... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | 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-017722 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Where appropriate, name the theorem you use: 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 criter... | [
{
"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-017723 | Linear Algebra: Minimal Polynomial Criterion | 9 | Indicate where a theorem is used: Consider the real matrix
$$A=\begin{pmatrix}12&1\\0&12\end{pmatrix}.$$
(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: $\\boxed{\\text{No}$.\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... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{No}$.) |
math-017724 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Carefully track domains: Consider the real matrix
$$A=\begin{pmatrix}-12&1\\0&-11\end{pmatrix}.$$
(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 justif... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-017725 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | Show all reasoning: Consider the real matrix
$$A=\begin{pmatrix}3&1\\0&7\end{pmatrix}.$$
(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 m... | [
{
"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-017726 | Linear Algebra: Minimal Polynomial Criterion | 9 | Exercise: Consider the real matrix
$$A=\begin{pmatrix}10&1\\0&10\end{pmatrix}.$$
(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 expl... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustne... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{No}$.) |
math-017727 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Give reasoning, not just computation: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&0&-3\\4&2&-5\\4&-1&-5\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two metho... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{6}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",
... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-017728 | Linear Algebra: Determinants — Cross-Validation | 9 | Do not skip justification steps: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&6&2\\2&4&6\\-3&-3&3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-12}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Takeaway: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-017729 | Matrix Theory: Determinant Properties | 9 | Start by stating any domain restrictions: Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&0&-5\\3&-1&1\\2&-1&-3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two ... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-19}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-19}$.) |
math-017730 | Linear Algebra: Minimal Polynomial Criterion | 9 | Where appropriate, name the theorem you use: Consider the real matrix
$$A=\begin{pmatrix}17&1\\0&-16\end{pmatrix}.$$
(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 crit... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-017731 | Linear Algebra: Minimal Polynomial Criterion | 9 | Work this out carefully: Consider the real matrix
$$A=\begin{pmatrix}8&1\\0&-2\end{pmatrix}.$$
(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 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": "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-017732 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Use two approaches if possible: Consider the real matrix
$$A=\begin{pmatrix}-10&1\\0&-10\end{pmatrix}.$$
(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": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustne... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{No}$.) |
math-017733 | Linear Algebra: Determinants — Row Operations | 9 | Warm-up: Compute the determinant of the matrix
$$A=\begin{pmatrix}0&6&0\\0&-2&-4\\-2&3&-1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.
When using ro... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{48}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",
... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{48}$.) |
math-017734 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | Provide both a computational and a conceptual explanation: 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 usin... | [
{
"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-017735 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | Start by stating any domain restrictions: Consider the real matrix
$$A=\begin{pmatrix}18&1\\0&18\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterio... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"ro... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | 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-017736 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Determine the requested value: Consider the real matrix
$$A=\begin{pmatrix}6&1\\0&-6\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your jus... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "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-017737 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Indicate where a theorem is used: Consider the real matrix
$$A=\begin{pmatrix}3&1\\0&3\end{pmatrix}.$$
(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 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: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"ro... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | 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-017738 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | Use two approaches if possible: Consider the real matrix
$$A=\begin{pmatrix}-11&1\\0&-11\end{pmatrix}.$$
(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... | [
{
"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: $\\boxed{\\text{No}$.\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... | 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-017739 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Start by stating any domain restrictions: Consider the real matrix
$$A=\begin{pmatrix}-7&1\\0&-7\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterio... | [
{
"method_name": "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... | 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-017740 | Linear Algebra: Determinants — Row Operations | 9 | Give a theorem-based solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&-4&6\\-3&-1&0\\-4&-4&0\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods mus... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{48}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yi... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{48}$.) |
math-017741 | Matrix Theory: Determinant Properties | 9 | Carefully track domains: Compute the determinant of the matrix
$$A=\begin{pmatrix}0&-6&-5\\-3&-4&1\\-5&6&6\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agre... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{112}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-017742 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | Solve and sanity-check: 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 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": "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-017743 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Give an answer and a quick verification: Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&1&0\\3&-5&-4\\-2&0&-1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two m... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-19}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-19}$.) |
math-017744 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Keep the final answer in boxed form: 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.
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: $\\boxed{\\text{No}$.\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). (Here the result is $\boxed{\text{No}$.) |
math-017745 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Complete the analysis: Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&4&5\\-5&4&3\\6&4&1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.
W... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-80}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-80}$.) |
math-017746 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | Use two approaches if possible: Consider the real matrix
$$A=\begin{pmatrix}9&1\\0&9\end{pmatrix}.$$
(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 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: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the s... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | 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-017747 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | State any required conditions first: Consider the real matrix
$$A=\begin{pmatrix}-18&1\\0&-18\end{pmatrix}.$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"ro... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | 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-017748 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Question: Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&-2&2\\0&5&2\\-3&-1&4\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.
When using r... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-90}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-017749 | Linear Algebra: Determinants — Row Operations | 9 | State any required conditions first: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&2&6\\4&-4&0\\-6&-1&5\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-228}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number."... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-228}$.) |
math-017750 | Linear Algebra: Minimal Polynomial Criterion | 9 | Solve (and briefly cross-validate): Consider the real matrix
$$A=\begin{pmatrix}6&1\\0&6\end{pmatrix}.$$
(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": "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... | 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-017751 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Determine the requested value: Compute the determinant of the matrix
$$A=\begin{pmatrix}-2&-4&-3\\-6&-3&-1\\-3&6&0\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods m... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{111}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same num... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{111}$.) |
math-017752 | Matrix Theory: Determinant Properties | 9 | Give a fully justified solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}-3&1&-4\\6&6&-1\\6&-1&-1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods m... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{189}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{189}$.) |
math-017753 | Linear Algebra: Determinants — Cross-Validation | 9 | Complete the analysis: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&2&-5\\-1&1&6\\-3&6&0\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.
... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-237}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must ... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Takeaway: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-017754 | Linear Algebra: Determinants — Row Operations | 9 | Provide both a computational and a conceptual explanation: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&5&5\\-6&6&-1\\2&1&-1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly expla... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-140}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number."... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Takeaway: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-140}$.) |
math-017755 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Write the solution set clearly: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&2&5\\1&-4&2\\3&5&-2\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{103}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{103}$.) |
math-017756 | Linear Algebra: Determinants — Row Operations | 9 | Give an answer and a quick verification: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&6&6\\-1&-2&-6\\-6&0&-3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two m... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{138}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same num... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-017757 | Linear Algebra: Minimal Polynomial Criterion | 9 | Solve and then verify: Consider the real matrix
$$A=\begin{pmatrix}-1&1\\0&-7\end{pmatrix}.$$
(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 justificat... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | 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-017758 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Solve and include a self-check: Consider the real matrix
$$A=\begin{pmatrix}-6&1\\0&12\end{pmatrix}.$$
(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 j... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | 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-017759 | Linear Algebra: Determinants — Cross-Validation | 9 | Where appropriate, name the theorem you use: Compute the determinant of the matrix
$$A=\begin{pmatrix}-5&-4&5\\3&4&0\\1&-3&-1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the tw... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-57}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both metho... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-57}$.) |
math-017760 | Matrix Theory: Determinant Properties | 9 | Work carefully and justify each inference: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&-5&5\\6&-3&-5\\0&-5&-3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-256}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number."... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-017761 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Try to avoid pattern-matching; explain why: Consider the real matrix
$$A=\begin{pmatrix}16&1\\0&-19\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named 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... | 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-017762 | Linear Algebra: Determinants — Row Operations | 9 | Answer with a short justification: Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&-5&-4\\6&-6&-1\\3&0&-6\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two method... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-453}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same nu... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-017763 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | Give a theorem-based solution: Consider the real matrix
$$A=\begin{pmatrix}20&1\\0&-7\end{pmatrix}.$$
(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 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": "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-017764 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Start by stating any domain restrictions: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&-3&-6\\-1&0&-6\\3&0&3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two m... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{45}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yi... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{45}$.) |
math-017765 | Matrix Theory: Determinant Properties | 9 | Question: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&-1&6\\0&2&-2\\2&3&3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.
When using row... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{16}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both method... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{16}$.) |
math-017766 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | Do not skip justification steps: Consider the real matrix
$$A=\begin{pmatrix}-1&1\\0&18\end{pmatrix}.$$
(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 ... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-017767 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Do not skip justification steps: Consider the real matrix
$$A=\begin{pmatrix}-3&1\\0&11\end{pmatrix}.$$
(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 ... | [
{
"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-017768 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | State any required conditions first: Consider the real matrix
$$A=\begin{pmatrix}2&1\\0&-1\end{pmatrix}.$$
(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.
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": "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-017769 | Linear Algebra: Minimal Polynomial Criterion | 9 | Give an answer and a quick verification: Consider the real matrix
$$A=\begin{pmatrix}-20&1\\0&-20\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named 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": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"ro... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | 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 $\boxed{\text{No}$.) |
math-017770 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | Keep the final answer in boxed form: Consider the real matrix
$$A=\begin{pmatrix}-7&1\\0&4\end{pmatrix}.$$
(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": "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-017771 | Linear Algebra: Determinants — Row Operations | 9 | Keep the final answer in boxed form: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&0&-5\\-6&-2&5\\5&-5&1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two method... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-131}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same nu... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-017772 | Linear Algebra: Determinants — Row Operations | 9 | Indicate where a theorem is used: Compute the determinant of the matrix
$$A=\begin{pmatrix}-3&-4&6\\4&-2&-4\\5&0&-1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods ... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{118}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{118}$.) |
math-017773 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Give an answer and a quick verification: Consider the real matrix
$$A=\begin{pmatrix}5&1\\0&5\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
... | [
{
"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: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"ro... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | 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 $\boxed{\text{No}$.) |
math-017774 | Linear Algebra: Determinants — Cross-Validation | 9 | Complete the analysis: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&1&2\\4&-3&-5\\-2&-1&-2\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{2}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yie... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{2}$.) |
math-017775 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Track quantifiers carefully: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&4&-6\\-6&5&-4\\1&1&6\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must ag... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{330}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{330}$.) |
math-017776 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Compute the requested quantity: Consider the real matrix
$$A=\begin{pmatrix}7&1\\0&-12\end{pmatrix}.$$
(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 j... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | 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-017777 | Linear Algebra: Determinants — Cross-Validation | 9 | Indicate where a theorem is used: Compute the determinant of the matrix
$$A=\begin{pmatrix}-3&-2&-2\\3&-2&4\\0&-1&-6\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-78}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-017778 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Do not skip justification steps: Compute the determinant of the matrix
$$A=\begin{pmatrix}0&-5&-4\\2&1&-6\\-1&3&4\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods mu... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-18}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-18}$.) |
math-017779 | Linear Algebra: Determinants — Row Operations | 9 | Do not skip justification steps: Compute the determinant of the matrix
$$A=\begin{pmatrix}0&-1&6\\-4&-3&0\\5&6&5\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods mus... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-74}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Takeaway: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-017780 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Answer with a short justification: Consider the real matrix
$$A=\begin{pmatrix}-15&1\\0&-14\end{pmatrix}.$$
(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.
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": "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-017781 | Linear Algebra: Minimal Polynomial Criterion | 9 | Challenge: Consider the real matrix
$$A=\begin{pmatrix}1&1\\0&-10\end{pmatrix}.$$
(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 exp... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-017782 | Matrix Theory: Determinant Properties | 9 | Carefully track domains: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&1&0\\5&4&6\\-4&2&-4\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.
... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-32}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-32}$.) |
math-017783 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Provide both a computational and a conceptual explanation: Consider the real matrix
$$A=\begin{pmatrix}-3&1\\0&-3\end{pmatrix}.$$
(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... | [
{
"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-017784 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Find the exact value: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&-4&-4\\1&6&-3\\2&-5&-4\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.
... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-158}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must ... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-158}$.) |
math-017785 | Linear Algebra: Determinants — Row Operations | 9 | Try to avoid pattern-matching; explain why: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&4&6\\-4&-2&4\\-2&0&3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two ... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-32}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both metho... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-017786 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Where appropriate, name the theorem you use: Consider the real matrix
$$A=\begin{pmatrix}11&1\\0&11\end{pmatrix}.$$
(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 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: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the s... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | 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-017787 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Indicate where a theorem is used: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&6&3\\-5&-3&3\\-2&-2&-3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods ... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-99}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-99}$.) |
math-017788 | Linear Algebra: Minimal Polynomial Criterion | 9 | Give an answer and a quick verification: Consider the real matrix
$$A=\begin{pmatrix}-13&1\\0&-9\end{pmatrix}.$$
(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 criterio... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\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-017789 | Matrix Theory: Determinant Properties | 9 | Work this out carefully: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&-4&-2\\-2&-1&-1\\3&-4&6\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agr... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-118}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both meth... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-118}$.) |
math-017790 | Matrix Theory: Determinant Properties | 9 | Track quantifiers carefully: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&1&-4\\-3&4&-4\\-5&3&1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must ... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-37}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both metho... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-37}$.) |
math-017791 | Matrix Theory: Determinant Properties | 9 | Question: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&5&-2\\2&5&4\\-5&-6&2\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.
When using ro... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-78}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Takeaway: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-017792 | Linear Algebra: Determinants — Row Operations | 9 | Track quantifiers carefully: Compute the determinant of the matrix
$$A=\begin{pmatrix}0&-6&-1\\5&-1&6\\-1&-3&3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must ... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{142}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{142}$.) |
math-017793 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Give reasoning, not just computation: Consider the real matrix
$$A=\begin{pmatrix}-7&1\\0&-19\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
... | [
{
"method_name": "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-017794 | Linear Algebra: Minimal Polynomial Criterion | 9 | Work carefully and justify each inference: Consider the real matrix
$$A=\begin{pmatrix}-13&1\\0&-13\end{pmatrix}.$$
(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": "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: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"ro... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{No}$.) |
math-017795 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Exercise: Consider the real matrix
$$A=\begin{pmatrix}-12&1\\0&-12\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must 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: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustne... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{No}$.) |
math-017796 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Where appropriate, name the theorem you use: Consider the real matrix
$$A=\begin{pmatrix}4&1\\0&-15\end{pmatrix}.$$
(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 crite... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-017797 | Linear Algebra: Minimal Polynomial Criterion | 9 | Explain why your operations are valid: 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.
... | [
{
"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). (Here the result is $\boxed{\text{Yes}$.) |
math-017798 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Explain why your operations are valid: Consider the real matrix
$$A=\begin{pmatrix}20&1\\0&-5\end{pmatrix}.$$
(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.
... | [
{
"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-017799 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Solve and include a self-check: Consider the real matrix
$$A=\begin{pmatrix}-17&1\\0&20\end{pmatrix}.$$
(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": "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-017800 | Linear Algebra: Minimal Polynomial Criterion | 9 | Proceed methodically: Consider the real matrix
$$A=\begin{pmatrix}19&1\\0&19\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justificati... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"ro... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | 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 $\boxed{\text{No}$.) |
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