id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-017801 | Linear Algebra: Determinants — Row Operations | 9 | Determine the requested value: Compute the determinant of the matrix
$$A=\begin{pmatrix}-5&5&-5\\5&1&5\\-5&-2&0\end{pmatrix}.$$
(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... | [
{
"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: $\\boxed{-150}$.\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... | 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 $\boxed{-150}$.) |
math-017802 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | State any required conditions first: Consider the real matrix
$$A=\begin{pmatrix}-5&1\\0&12\end{pmatrix}.$$
(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.
Y... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "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-017803 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Write the solution set clearly: Consider the real matrix
$$A=\begin{pmatrix}-12&1\\0&-16\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-017804 | Linear Algebra: Determinants — Row Operations | 9 | Complete the analysis: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&-4&-6\\2&-3&2\\-6&5&3\end{pmatrix}.$$
(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 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: $\\boxed{63}$.\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 $\boxed{63}$.) |
math-017805 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Explain each transformation: Consider the real matrix
$$A=\begin{pmatrix}-16&1\\0&-20\end{pmatrix}.$$
(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 ju... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-017806 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Work carefully and justify each inference: Consider the real matrix
$$A=\begin{pmatrix}10&1\\0&20\end{pmatrix}.$$
(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": "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-017807 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | Track quantifiers carefully: Consider the real matrix
$$A=\begin{pmatrix}8&1\\0&8\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your 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: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"ro... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{No}$.) |
math-017808 | Matrix Theory: Determinant Properties | 9 | Explain each transformation: Compute the determinant of the matrix
$$A=\begin{pmatrix}-3&4&5\\-1&0&6\\-6&4&-4\end{pmatrix}.$$
(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": "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: $\\boxed{-108}$.\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-017809 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Checkpoint: Compute the determinant of the matrix
$$A=\begin{pmatrix}-4&0&-2\\0&-4&3\\-5&1&2\end{pmatrix}.$$
(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.
When using... | [
{
"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: $\\boxed{84}$.\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... | 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-017810 | Linear Algebra: Minimal Polynomial Criterion | 9 | Solve and justify each step: Consider the real matrix
$$A=\begin{pmatrix}20&1\\0&19\end{pmatrix}.$$
(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 just... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-017811 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | Indicate where a theorem is used: Consider the real matrix
$$A=\begin{pmatrix}-4&1\\0&-4\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "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-017812 | Linear Algebra: Determinants — Row Operations | 9 | Work carefully and justify each inference: Compute the determinant of the matrix
$$A=\begin{pmatrix}0&3&4\\-4&4&6\\-3&-6&-3\end{pmatrix}.$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{54}$.\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 numb... | [
{
"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-017813 | Linear Algebra: Minimal Polynomial Criterion | 9 | Solve with verification: 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 criterion.
Your justific... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{No}$.\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... | 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-017814 | Matrix Theory: Determinant Properties | 9 | Work carefully and justify each inference: Compute the determinant of the matrix
$$A=\begin{pmatrix}5&2&6\\3&0&-2\\-1&-5&4\end{pmatrix}.$$
(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 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": "Both approaches agree after simplification. Final answer: $\\boxed{-160}$.\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 $\boxed{-160}$.) |
math-017815 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Use two approaches if possible: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&4&4\\0&2&-1\\-2&-2&2\end{pmatrix}.$$
(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": "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: $\\boxed{28}$.\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 $\boxed{28}$.) |
math-017816 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Complete the analysis: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&-1&-2\\1&2&2\\-3&1&-6\end{pmatrix}.$$
(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: $\\boxed{-28}$.\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 $\boxed{-28}$.) |
math-017817 | Linear Algebra: Determinants — Row Operations | 9 | Question: Compute the determinant of the matrix
$$A=\begin{pmatrix}-3&4&0\\-4&-5&3\\-1&0&0\end{pmatrix}.$$
(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.
When using r... | [
{
"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... | 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-017818 | Matrix Theory: Determinant Properties | 9 | Indicate where a theorem is used: Compute the determinant of the matrix
$$A=\begin{pmatrix}-2&-2&1\\5&4&2\\1&-6&-3\end{pmatrix}.$$
(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 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: $\\boxed{-68}$.\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 $\boxed{-68}$.) |
math-017819 | Linear Algebra: Determinants — Cross-Validation | 9 | Answer using clear logical steps: Compute the determinant of the matrix
$$A=\begin{pmatrix}5&6&-5\\6&-3&6\\-6&-4&-4\end{pmatrix}.$$
(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 ... | [
{
"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: $\\boxed{318}$.\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-017820 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Question: 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 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": "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-017821 | Linear Algebra: Determinants — Row Operations | 9 | Track units/moduli carefully: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&-1&-3\\4&0&0\\4&6&6\end{pmatrix}.$$
(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 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": "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 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-017822 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Show all reasoning: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&2&2\\6&-5&-1\\0&2&6\end{pmatrix}.$$
(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": "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: $\\boxed{-216}$.\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... | 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 $\boxed{-216}$.) |
math-017823 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Start by stating any domain restrictions: Consider the real matrix
$$A=\begin{pmatrix}-8&1\\0&-5\end{pmatrix}.$$
(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 criterio... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-017824 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Start by stating any domain restrictions: Consider the real matrix
$$A=\begin{pmatrix}13&1\\0&-4\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named 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... | 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-017825 | Matrix Theory: Determinant Properties | 9 | Give a theorem-based solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&-3&6\\1&4&2\\-6&5&-6\end{pmatrix}.$$
(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... | [
{
"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: $\\boxed{226}$.\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 $\boxed{226}$.) |
math-017826 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Solve and then verify: Consider the real matrix
$$A=\begin{pmatrix}3&1\\0&-2\end{pmatrix}.$$
(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 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": "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-017827 | Linear Algebra: Determinants — Cross-Validation | 9 | Solve and then verify: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&-5&-3\\-4&6&5\\1&3&-5\end{pmatrix}.$$
(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": "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: $\\boxed{39}$.\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... | 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-017828 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Explain why your operations are valid: Consider the real matrix
$$A=\begin{pmatrix}4&1\\0&4\end{pmatrix}.$$
(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.
Y... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the s... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{No}$.) |
math-017829 | Linear Algebra: Minimal Polynomial Criterion | 9 | Try to avoid pattern-matching; explain why: Consider the real matrix
$$A=\begin{pmatrix}-8&1\\0&-8\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named 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": "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-017830 | Linear Algebra: Minimal Polynomial Criterion | 9 | Solve and then verify: 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.
Your justifica... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "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-017831 | Linear Algebra: Determinants — Row Operations | 9 | Determine the requested value: Compute the determinant of the matrix
$$A=\begin{pmatrix}-3&-6&-6\\-6&6&6\\6&5&2\end{pmatrix}.$$
(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": "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: $\\boxed{162}$.\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... | 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-017832 | Linear Algebra: Determinants — Cross-Validation | 9 | Provide a rigorous solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&-1&4\\-1&6&-2\\-2&2&-3\end{pmatrix}.$$
(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 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": "The two methods are consistent and must coincide. Final answer: $\\boxed{123}$.\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. (Here the result is $\boxed{123}$.) |
math-017833 | Linear Algebra: Minimal Polynomial Criterion | 9 | State any required conditions first: 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.
... | [
{
"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... | 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-017834 | Matrix Theory: Determinant Properties | 9 | Find the exact value: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&4&-2\\2&5&6\\-3&-2&0\end{pmatrix}.$$
(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.
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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-46}$.\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... | 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 $\boxed{-46}$.) |
math-017835 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | Complete the analysis: Consider the real matrix
$$A=\begin{pmatrix}-13&1\\0&-2\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justifica... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "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-017836 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | Proceed methodically: 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 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": "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-017837 | Linear Algebra: Determinants — Row Operations | 9 | Challenge: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&-2&3\\6&6&2\\-3&2&-2\end{pmatrix}.$$
(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 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": "The two methods are consistent and must coincide. Final answer: $\\boxed{30}$.\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 numb... | [
{
"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 $\boxed{30}$.) |
math-017838 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Show all reasoning: Consider the real matrix
$$A=\begin{pmatrix}12&1\\0&16\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification... | [
{
"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-017839 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Answer with a short justification: Compute the determinant of the matrix
$$A=\begin{pmatrix}5&-4&-1\\-1&1&5\\3&6&4\end{pmatrix}.$$
(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 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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-197}$.\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... | 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-017840 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Try to avoid pattern-matching; explain why: Consider the real matrix
$$A=\begin{pmatrix}-17&1\\0&-14\end{pmatrix}.$$
(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 crit... | [
{
"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-017841 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Indicate where a theorem is used: 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 ... | [
{
"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-017842 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Proceed methodically: Consider the real matrix
$$A=\begin{pmatrix}16&1\\0&16\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justificati... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "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-017843 | Matrix Theory: Determinant Properties | 9 | Task: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&4&-4\\6&-5&1\\1&5&4\end{pmatrix}.$$
(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.
When using row red... | [
{
"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: $\\boxed{-307}$.\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... | 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 $\boxed{-307}$.) |
math-017844 | Linear Algebra: Minimal Polynomial Criterion | 9 | Determine the requested value: 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 jus... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | 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-017845 | Matrix Theory: Determinant Properties | 9 | Explain what is being counted/optimized: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&-5&-3\\-6&0&-1\\3&1&3\end{pmatrix}.$$
(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 me... | [
{
"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: $\\boxed{-51}$.\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-017846 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Challenge: Consider the real matrix
$$A=\begin{pmatrix}14&1\\0&16\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must exp... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-017847 | Linear Algebra: Determinants — Cross-Validation | 9 | Indicate where a theorem is used: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&-4&6\\1&-6&-4\\6&-4&5\end{pmatrix}.$$
(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 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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{170}$.\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. (Here the result is $\boxed{170}$.) |
math-017848 | Linear Algebra: Determinants — Cross-Validation | 9 | Answer with a short justification: Compute the determinant of the matrix
$$A=\begin{pmatrix}-5&5&5\\6&-1&3\\0&4&3\end{pmatrix}.$$
(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 mu... | [
{
"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: $\\boxed{105}$.\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. |
math-017849 | Linear Algebra: Determinants — Row Operations | 9 | Compute the requested quantity: Compute the determinant of the matrix
$$A=\begin{pmatrix}5&-1&-6\\-1&-3&3\\1&1&6\end{pmatrix}.$$
(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 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: $\\boxed{-126}$.\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-017850 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Proceed methodically: Compute the determinant of the matrix
$$A=\begin{pmatrix}5&-5&-4\\1&-5&3\\-5&0&5\end{pmatrix}.$$
(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": "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: $\\boxed{75}$.\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 numb... | [
{
"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 $\boxed{75}$.) |
math-017851 | Linear Algebra: Determinants — Cross-Validation | 9 | Complete the analysis: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&0&-4\\1&-4&-1\\4&-5&-5\end{pmatrix}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. 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 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 $\boxed{16}$.) |
math-017852 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Explain what is being counted/optimized: Consider the real matrix
$$A=\begin{pmatrix}14&1\\0&5\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-017853 | Linear Algebra: Minimal Polynomial Criterion | 9 | Warm-up: Consider the real matrix
$$A=\begin{pmatrix}-9&1\\0&13\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must expli... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | 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-017854 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Provide a rigorous solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&5&-3\\0&-4&5\\-5&-3&2\end{pmatrix}.$$
(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 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: $\\boxed{-37}$.\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. |
math-017855 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | Keep the final answer in boxed form: Consider the real matrix
$$A=\begin{pmatrix}9&1\\0&17\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
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... | 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-017856 | Linear Algebra: Minimal Polynomial Criterion | 9 | Compute the requested quantity: Consider the real matrix
$$A=\begin{pmatrix}-18&1\\0&20\end{pmatrix}.$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the ... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-017857 | Linear Algebra: Determinants — Row Operations | 9 | Prompt: Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&-5&-3\\1&-1&-3\\2&3&6\end{pmatrix}.$$
(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 using ro... | [
{
"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: $\\boxed{27}$.\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... | 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 $\boxed{27}$.) |
math-017858 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Checkpoint: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&4&1\\-1&4&2\\-1&-1&2\end{pmatrix}.$$
(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.
When using ... | [
{
"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: $\\boxed{45}$.\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... | 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-017859 | Linear Algebra: Minimal Polynomial Criterion | 9 | Work this out carefully: Consider the real matrix
$$A=\begin{pmatrix}2&1\\0&-9\end{pmatrix}.$$
(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 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). (Here the result is $\boxed{\text{Yes}$.) |
math-017860 | Linear Algebra: Determinants — Cross-Validation | 9 | Warm-up: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&2&0\\6&2&1\\1&2&-3\end{pmatrix}.$$
(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 agree.
When using row 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: $\\boxed{6}$.\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... | 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-017861 | Matrix Theory: Determinant Properties | 9 | Write the solution set clearly: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&5&6\\5&-5&0\\5&2&-4\end{pmatrix}.$$
(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 ... | [
{
"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: $\\boxed{390}$.\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 $\boxed{390}$.) |
math-017862 | Linear Algebra: Determinants — Row Operations | 9 | Solve and justify each step: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&5&-5\\6&-5&1\\-6&6&6\end{pmatrix}.$$
(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 ag... | [
{
"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: $\\boxed{-276}$.\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... | 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-017863 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Start by stating any domain restrictions: Consider the real matrix
$$A=\begin{pmatrix}-17&1\\0&18\end{pmatrix}.$$
(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 criteri... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-017864 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Work carefully and justify each inference: Consider the real matrix
$$A=\begin{pmatrix}-1&1\\0&19\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criteri... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | 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-017865 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | Track quantifiers carefully: Consider the real matrix
$$A=\begin{pmatrix}-3&1\\0&16\end{pmatrix}.$$
(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 just... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-017866 | Linear Algebra: Determinants — Row Operations | 9 | Provide a rigorous solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}-4&-1&5\\-4&1&4\\-3&0&-5\end{pmatrix}.$$
(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 ... | [
{
"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: $\\boxed{67}$.\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 $\boxed{67}$.) |
math-017867 | Linear Algebra: Determinants — Row Operations | 9 | Carefully track domains: Compute the determinant of the matrix
$$A=\begin{pmatrix}-4&4&-3\\-5&3&-2\\4&-5&-2\end{pmatrix}.$$
(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 agr... | [
{
"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: $\\boxed{-47}$.\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. |
math-017868 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Show all reasoning: Compute the determinant of the matrix
$$A=\begin{pmatrix}-4&4&-3\\3&3&-5\\-4&0&1\end{pmatrix}.$$
(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 must agree.
Wh... | [
{
"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: $\\boxed{20}$.\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... | 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 $\boxed{20}$.) |
math-017869 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Proceed methodically: Compute the determinant of the matrix
$$A=\begin{pmatrix}-5&-6&3\\0&-3&-6\\2&1&-5\end{pmatrix}.$$
(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 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... | 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 $\boxed{-15}$.) |
math-017870 | Linear Algebra: Determinants — Row Operations | 9 | Indicate where a theorem is used: Compute the determinant of the matrix
$$A=\begin{pmatrix}-5&5&-1\\5&-3&6\\5&3&-3\end{pmatrix}.$$
(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 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": "Both approaches agree after simplification. Final answer: $\\boxed{240}$.\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 $\boxed{240}$.) |
math-017871 | Linear Algebra: Minimal Polynomial Criterion | 9 | Warm-up: Consider the real matrix
$$A=\begin{pmatrix}-1&1\\0&6\end{pmatrix}.$$
(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 explic... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-017872 | Linear Algebra: Determinants — Cross-Validation | 9 | Determine the requested value: Compute the determinant of the matrix
$$A=\begin{pmatrix}5&-1&-3\\4&-6&2\\-3&4&0\end{pmatrix}.$$
(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 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: $\\boxed{-28}$.\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. (Here the result is $\boxed{-28}$.) |
math-017873 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Make each step logically reversible (or explain if not): Consider the real matrix
$$A=\begin{pmatrix}-20&1\\0&14\end{pmatrix}.$$
(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 ... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-017874 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | Give an answer and a quick verification: Consider the real matrix
$$A=\begin{pmatrix}-1&1\\0&-20\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterio... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-017875 | Linear Algebra: Jordan Form Intuition (2×2) | 9 | Find the exact value: Consider the real matrix
$$A=\begin{pmatrix}6&1\\0&13\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justificatio... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-017876 | Linear Algebra: Determinants — Row Operations | 9 | Checkpoint: Compute the determinant of the matrix
$$A=\begin{pmatrix}0&3&-6\\4&0&6\\-3&-1&-5\end{pmatrix}.$$
(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 must agree.
When using... | [
{
"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: $\\boxed{30}$.\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 numb... | [
{
"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 $\boxed{30}$.) |
math-017877 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Do not skip justification steps: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&2&-3\\5&1&2\\-4&-6&-4\end{pmatrix}.$$
(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 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": "The two methods are consistent and must coincide. Final answer: $\\boxed{150}$.\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... | 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-017878 | Matrix Theory: Determinant Properties | 9 | Problem: Compute the determinant of the matrix
$$A=\begin{pmatrix}-2&5&-1\\-1&2&-2\\-5&-2&5\end{pmatrix}.$$
(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.
When using ... | [
{
"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: $\\boxed{51}$.\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. |
math-017879 | Linear Algebra: Determinants — Cross-Validation | 9 | Proceed methodically: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&3&0\\6&6&4\\-2&5&5\end{pmatrix}.$$
(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.
Whe... | [
{
"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-017880 | Linear Algebra: Minimal Polynomial Criterion | 9 | Prompt: Consider the real matrix
$$A=\begin{pmatrix}20&1\\0&20\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must explic... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{No}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the s... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Remember: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{No}$.) |
math-017881 | Matrix Theory: Determinant Properties | 9 | Give a fully justified solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&2&3\\2&-3&-5\\5&2&6\end{pmatrix}.$$
(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... | [
{
"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: $\\boxed{-33}$.\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 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 $\boxed{-33}$.) |
math-017882 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Give a theorem-based solution: Consider the real matrix
$$A=\begin{pmatrix}7&1\\0&-13\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your ju... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | 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-017883 | Linear Algebra: Minimal Polynomial Criterion | 9 | Track quantifiers carefully: Consider the real matrix
$$A=\begin{pmatrix}6&1\\0&9\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justif... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "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-017884 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Complete the analysis: Consider the real matrix
$$A=\begin{pmatrix}-1&1\\0&-1\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named 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: $\\boxed{\\text{No}$.\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... | 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-017885 | Linear Algebra: Minimal Polynomial Criterion | 9 | Explain what is being counted/optimized: Consider the real matrix
$$A=\begin{pmatrix}-2&1\\0&-2\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "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-017886 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Question: Compute the determinant of the matrix
$$A=\begin{pmatrix}-4&-6&4\\3&-3&6\\5&-2&4\end{pmatrix}.$$
(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.
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: $\\boxed{-72}$.\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 $\boxed{-72}$.) |
math-017887 | Linear Algebra: Determinants — Row Operations | 9 | Solve and then verify: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&6&0\\4&-3&-3\\2&1&-4\end{pmatrix}.$$
(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 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": "The two methods are consistent and must coincide. 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 yield the same numb... | [
{
"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 $\boxed{45}$.) |
math-017888 | Linear Algebra: Determinants — Row Operations | 9 | Answer with a short justification: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&4&-2\\-6&3&-2\\-2&-1&5\end{pmatrix}.$$
(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": "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: $\\boxed{125}$.\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 $\boxed{125}$.) |
math-017889 | Linear Algebra: Determinants — Row Operations | 9 | Track quantifiers carefully: Compute the determinant of the matrix
$$A=\begin{pmatrix}-3&-6&-4\\-1&-4&-5\\-5&-2&-6\end{pmatrix}.$$
(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 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: $\\boxed{-84}$.\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. |
math-017890 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Answer using clear logical steps: Compute the determinant of the matrix
$$A=\begin{pmatrix}-5&-1&-5\\-5&-4&-1\\1&2&3\end{pmatrix}.$$
(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... | [
{
"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: $\\boxed{66}$.\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... | 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 $\boxed{66}$.) |
math-017891 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Do not skip justification steps: Compute the determinant of the matrix
$$A=\begin{pmatrix}5&3&3\\-4&-2&2\\-6&1&2\end{pmatrix}.$$
(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 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: $\\boxed{-90}$.\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 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 $\boxed{-90}$.) |
math-017892 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | Explain what is being counted/optimized: Consider the real matrix
$$A=\begin{pmatrix}-18&1\\0&-9\end{pmatrix}.$$
(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": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"robustn... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Core principle: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). (Here the result is $\boxed{\text{Yes}$.) |
math-017893 | Matrix Theory: Determinant Properties | 9 | Carefully track domains: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&3&5\\-1&2&-2\\-1&6&0\end{pmatrix}.$$
(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": "Consistency verification shows both paths yield the identical boxed result. 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 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... | 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-017894 | Linear Algebra: Minimal Polynomial Criterion | 9 | Start by stating any domain restrictions: Consider the real matrix
$$A=\begin{pmatrix}-7&1\\0&-3\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterio... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "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-017895 | Linear Algebra: Algebraic vs Geometric Multiplicity | 9 | Solve and sanity-check: Consider the real matrix
$$A=\begin{pmatrix}16&1\\0&0\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your 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... | 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-017896 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | Do not skip justification steps: 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... | [
{
"method_name": "Minimal Polynomial / Jordan Block",
"approach": "Diagonalizable over $\\mathbb{R}$ iff the minimal polynomial splits with no repeated linear factors; a nontrivial Jordan block for a repeated eigenvalue prevents diagonalization.",
"steps": [
"Step 1: If eigenvalues are distinct, t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/no answer.",
"r... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Takeaway: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-017897 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Problem: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&-2&-2\\3&4&3\\4&4&6\end{pmatrix}.$$
(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 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": "The two methods are consistent and must coincide. Final answer: $\\boxed{56}$.\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 numb... | [
{
"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 $\boxed{56}$.) |
math-017898 | Matrix Theory: Determinant Properties | 9 | Solve with verification: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&2&0\\6&-1&-6\\-4&-6&2\end{pmatrix}.$$
(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... | [
{
"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: $\\boxed{-14}$.\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 $\boxed{-14}$.) |
math-017899 | Linear Algebra: Diagonalizability — Eigenvectors | 9 | Problem: Consider the real matrix
$$A=\begin{pmatrix}-6&1\\0&-9\end{pmatrix}.$$
(a) Determine the eigenvalues and their algebraic multiplicities.
(b) Compute the dimension of each eigenspace.
(c) Decide whether $A$ is diagonalizable over $\mathbb{R}$, and justify using a named criterion.
Your justification must expli... | [
{
"method_name": "Eigenvectors vs Algebraic Multiplicity",
"approach": "A $2\\times2$ matrix is diagonalizable iff it has 2 linearly independent eigenvectors; equivalently each eigenvalue's geometric multiplicity equals its algebraic multiplicity.",
"steps": [
"Step 1: Compute $\\chi_A(\\lambda)=\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{Yes}$.\nThe eigenvector-count criterion and the minimal-polynomial/Jordan criterion are equivalent: both detect whether there is a full eigenbasis. They therefore necessarily agree on the same yes/... | [
{
"error_description": "Assumed distinct eigenvalues are necessary (not just sufficient) for diagonalizability.",
"why_plausible": "Many examples emphasize the distinct-eigenvalue shortcut.",
"why_wrong": "Matrices with repeated eigenvalues can still be diagonalizable if they have enough eigenvectors (e... | Key idea: Diagonalizability is about eigenvectors, not just eigenvalues: you need a full eigenbasis. Repeated eigenvalues force you to check eigenspace dimension (or minimal polynomial). |
math-017900 | Linear Algebra: Determinants — Cofactor Expansion | 9 | Be explicit about assumptions: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&-5&2\\5&-4&-4\\-4&1&-6\end{pmatrix}.$$
(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 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": "Cross-check: both derivations land on the same invariant quantity. 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 ... | [
{
"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 $\boxed{-140}$.) |
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