id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-013701 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Compute the requested quantity: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&0&-5\\2&-3&4\\-4&5&5\end{pmatrix}.$$
(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... | [
{
"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: $\\boxed{-200}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must ... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-200}$.) |
math-013702 | Linear Algebra: Determinants — Cross-Validation | 7 | Work carefully and justify each inference: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&-6&-4\\6&0&-4\\-6&6&-6\end{pmatrix}.$$
(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... | [
{
"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: $\\boxed{-480}$.\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 $\boxed{-480}$.) |
math-013703 | Linear Algebra: Determinants — Cross-Validation | 7 | Explain each transformation: Compute the determinant of the matrix
$$A=\begin{pmatrix}0&-4&-4\\5&-3&-2\\-3&3&-3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must... | [
{
"method_name": "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... | 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-013704 | Linear Algebra: Determinants — Row Operations | 7 | Solve (and briefly cross-validate): Compute the determinant of the matrix
$$A=\begin{pmatrix}4&-1&-4\\-3&3&0\\0&6&0\end{pmatrix}.$$
(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 ... | [
{
"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: $\\boxed{72}$.\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-013705 | Linear Algebra: Determinants — Cross-Validation | 7 | Solve (and briefly cross-validate): Compute the determinant of the matrix
$$A=\begin{pmatrix}5&-6&3\\-3&-6&2\\1&4&1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods ... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-118}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number."... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-013706 | Linear Algebra: Determinants — Cross-Validation | 7 | Answer using clear logical steps: Compute the determinant of the matrix
$$A=\begin{pmatrix}-5&-3&3\\4&-1&0\\-4&2&4\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods 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: $\\boxed{80}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must 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 $\boxed{80}$.) |
math-013707 | Matrix Theory: Determinant Properties | 7 | Solve (and briefly cross-validate): Compute the determinant of the matrix
$$A=\begin{pmatrix}1&6&-4\\0&5&-6\\-2&-1&-6\end{pmatrix}.$$
(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 method... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-4}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same 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... | 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-013708 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Work carefully and justify each inference: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&-2&0\\-1&-3&0\\-4&3&-4\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{80}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must 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-013709 | Matrix Theory: Determinant Properties | 7 | Show all reasoning: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&1&1\\-6&-4&-5\\-4&5&-2\end{pmatrix}.$$
(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 agree.
W... | [
{
"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: $\\boxed{61}$.\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 $\boxed{61}$.) |
math-013710 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Be explicit about assumptions: Compute the determinant of the matrix
$$A=\begin{pmatrix}-3&-5&-1\\3&2&2\\6&6&-2\end{pmatrix}.$$
(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... | [
{
"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... | 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-013711 | Matrix Theory: Determinant Properties | 7 | Start by stating any domain restrictions: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&5&2\\4&-6&6\\-4&-2&3\end{pmatrix}.$$
(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 me... | [
{
"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: $\\boxed{-262}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must ... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-013712 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Give a fully justified solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&-3&1\\-3&3&2\\1&1&6\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must... | [
{
"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: $\\boxed{-18}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-013713 | Linear Algebra: Determinants — Row Operations | 7 | Start by stating any domain restrictions: Compute the determinant of the matrix
$$A=\begin{pmatrix}-3&5&-4\\-2&6&6\\5&-2&1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two m... | [
{
"method_name": "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: $\\boxed{210}$.\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-013714 | Linear Algebra: Determinants — Cross-Validation | 7 | Where appropriate, name the theorem you use: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&1&-6\\-1&2&3\\1&-5&5\end{pmatrix}.$$
(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... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{140}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Takeaway: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-013715 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Derive the result step-by-step: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&5&5\\3&-1&-6\\-2&-6&0\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods 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: $\\boxed{-112}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must ... | [
{
"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 $\boxed{-112}$.) |
math-013716 | Linear Algebra: Determinants — Row Operations | 7 | Carefully track domains: Compute the determinant of the matrix
$$A=\begin{pmatrix}0&4&-3\\-1&-5&-4\\-2&4&0\end{pmatrix}.$$
(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 agre... | [
{
"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: $\\boxed{74}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both method... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{74}$.) |
math-013717 | Linear Algebra: Determinants — Row Operations | 7 | Explain why your operations are valid: Compute the determinant of the matrix
$$A=\begin{pmatrix}0&4&2\\-4&4&6\\3&1&2\end{pmatrix}.$$
(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... | [
{
"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: $\\boxed{72}$.\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 $\boxed{72}$.) |
math-013718 | Linear Algebra: Determinants — Cross-Validation | 7 | Derive the result step-by-step: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&5&-5\\-4&3&-3\\0&6&5\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods 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: $\\boxed{418}$.\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 $\boxed{418}$.) |
math-013719 | Matrix Theory: Determinant Properties | 7 | Do not skip justification steps: Compute the determinant of the matrix
$$A=\begin{pmatrix}0&-5&-5\\6&6&6\\5&1&-6\end{pmatrix}.$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{-210}$.\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 $\boxed{-210}$.) |
math-013720 | Linear Algebra: Determinants — Row Operations | 7 | Provide both a computational and a conceptual explanation: Compute the determinant of the matrix
$$A=\begin{pmatrix}-5&-3&-6\\-2&-4&-5\\0&0&-4\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly e... | [
{
"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: $\\boxed{-56}$.\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 y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-56}$.) |
math-013721 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Problem: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&1&-1\\4&-1&-3\\-4&-5&-2\end{pmatrix}.$$
(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 agree.
When using ... | [
{
"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: $\\boxed{5}$.\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 numbe... | [
{
"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-013722 | Matrix Theory: Determinant Properties | 7 | Carefully track domains: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&4&1\\-1&-6&4\\-3&-6&-6\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agre... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "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 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 $\boxed{276}$.) |
math-013723 | Linear Algebra: Determinants — Cross-Validation | 7 | State any required conditions first: Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&-3&6\\-1&-5&-5\\-1&-5&-6\end{pmatrix}.$$
(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 me... | [
{
"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: $\\boxed{-27}$.\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 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. |
math-013724 | Linear Algebra: Determinants — Row Operations | 7 | Challenge: Compute the determinant of the matrix
$$A=\begin{pmatrix}-4&4&3\\-5&6&-6\\6&0&-2\end{pmatrix}.$$
(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": "Both approaches agree after simplification. Final answer: $\\boxed{-244}$.\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-013725 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Give a fully justified solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&1&1\\-6&0&3\\1&6&-4\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must... | [
{
"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: $\\boxed{-165}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must ... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Takeaway: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-165}$.) |
math-013726 | Linear Algebra: Determinants — Cross-Validation | 7 | Solve with verification: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&2&6\\-2&3&2\\-3&-2&0\end{pmatrix}.$$
(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.... | [
{
"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: $\\boxed{90}$.\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 $\boxed{90}$.) |
math-013727 | Linear Algebra: Determinants — Row Operations | 7 | Write the solution set clearly: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&2&-4\\-5&-1&0\\3&5&-3\end{pmatrix}.$$
(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 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": "The two methods are consistent and must coincide. Final answer: $\\boxed{76}$.\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-013728 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Indicate where a theorem is used: Compute the determinant of the matrix
$$A=\begin{pmatrix}-2&0&-4\\0&-6&-3\\3&4&5\end{pmatrix}.$$
(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": "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: $\\boxed{-36}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-013729 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Exercise: Compute the determinant of the matrix
$$A=\begin{pmatrix}0&4&-1\\-6&2&-3\\1&-1&3\end{pmatrix}.$$
(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 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": "Cross-check: both derivations land on the same invariant quantity. 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 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 $\boxed{56}$.) |
math-013730 | Linear Algebra: Determinants — Cross-Validation | 7 | Make each step logically reversible (or explain if not): Compute the determinant of the matrix
$$A=\begin{pmatrix}-2&2&-6\\3&-2&1\\3&-1&2\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explai... | [
{
"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: $\\boxed{-18}$.\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 $\boxed{-18}$.) |
math-013731 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Carefully track domains: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&-2&-2\\0&2&-3\\-1&-6&4\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agre... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-70}$.\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. (Here the result is $\boxed{-70}$.) |
math-013732 | Linear Algebra: Determinants — Cross-Validation | 7 | Be explicit about assumptions: Compute the determinant of the matrix
$$A=\begin{pmatrix}-4&-3&6\\-2&1&3\\5&-1&-1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods 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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-65}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both 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. |
math-013733 | Linear Algebra: Determinants — Row Operations | 7 | Prompt: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&6&-2\\-2&-4&1\\2&4&1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.
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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-24}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-013734 | Matrix Theory: Determinant Properties | 7 | Prompt: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&6&-1\\-5&6&1\\-2&-3&3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.
When using row... | [
{
"method_name": "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: $\\boxed{114}$.\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-013735 | Linear Algebra: Determinants — Cross-Validation | 7 | Give an answer and a quick verification: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&0&0\\2&-5&3\\0&-1&2\end{pmatrix}.$$
(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 meth... | [
{
"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: $\\boxed{-7}$.\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. |
math-013736 | Linear Algebra: Determinants — Cross-Validation | 7 | Give reasoning, not just computation: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&0&-4\\-1&-3&6\\6&3&1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two method... | [
{
"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: $\\boxed{-102}$.\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-013737 | Matrix Theory: Determinant Properties | 7 | Find the exact value: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&4&-2\\1&-3&3\\-3&-5&3\end{pmatrix}.$$
(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": "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: $\\boxed{4}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yie... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | 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-013738 | Matrix Theory: Determinant Properties | 7 | Work carefully and justify each inference: Compute the determinant of the matrix
$$A=\begin{pmatrix}5&-4&-6\\-1&5&-6\\-2&1&4\end{pmatrix}.$$
(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... | [
{
"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: $\\boxed{12}$.\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 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 $\boxed{12}$.) |
math-013739 | Linear Algebra: Determinants — Cross-Validation | 7 | Complete the analysis: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&-3&4\\-4&0&3\\1&-3&-6\end{pmatrix}.$$
(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.
... | [
{
"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: $\\boxed{147}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-013740 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Explain each transformation: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&-1&5\\2&6&-2\\1&-4&-3\end{pmatrix}.$$
(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 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: $\\boxed{-48}$.\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... | 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-013741 | Linear Algebra: Determinants — Row Operations | 7 | Give an answer and a quick verification: Compute the determinant of the matrix
$$A=\begin{pmatrix}-2&3&5\\5&1&5\\0&3&-4\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two meth... | [
{
"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: $\\boxed{173}$.\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 $\boxed{173}$.) |
math-013742 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Solve and justify each step: Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&-3&3\\-2&3&3\\-3&-6&1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods 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: $\\boxed{-42}$.\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-013743 | Linear Algebra: Determinants — Cross-Validation | 7 | Explain why your operations are valid: Compute the determinant of the matrix
$$A=\begin{pmatrix}-3&-5&1\\1&6&-5\\-6&-5&2\end{pmatrix}.$$
(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 met... | [
{
"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: $\\boxed{-70}$.\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. |
math-013744 | Matrix Theory: Determinant Properties | 7 | Give reasoning, not just computation: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&-5&2\\3&-5&5\\-4&-3&-5\end{pmatrix}.$$
(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 meth... | [
{
"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: $\\boxed{127}$.\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. (Here the result is $\boxed{127}$.) |
math-013745 | Linear Algebra: Determinants — Cross-Validation | 7 | Use two approaches if possible: Compute the determinant of the matrix
$$A=\begin{pmatrix}-5&1&2\\-2&-4&-5\\-1&-4&3\end{pmatrix}.$$
(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 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: $\\boxed{179}$.\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-013746 | Matrix Theory: Determinant Properties | 7 | Answer using clear logical steps: Compute the determinant of the matrix
$$A=\begin{pmatrix}-2&-2&-2\\6&1&5\\0&4&5\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods 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: $\\boxed{42}$.\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 $\boxed{42}$.) |
math-013747 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Solve and justify each step: Compute the determinant of the matrix
$$A=\begin{pmatrix}-4&3&6\\2&2&5\\-5&3&-1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must 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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{95}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yi... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{95}$.) |
math-013748 | Linear Algebra: Determinants — Cross-Validation | 7 | Provide both a computational and a conceptual explanation: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&4&-1\\-3&-5&2\\-5&6&-6\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly exp... | [
{
"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: $\\boxed{-15}$.\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... | 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 $\boxed{-15}$.) |
math-013749 | Linear Algebra: Determinants — Cross-Validation | 7 | Where appropriate, name the theorem you use: Compute the determinant of the matrix
$$A=\begin{pmatrix}-2&-1&1\\3&-1&6\\5&-3&-1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the t... | [
{
"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: $\\boxed{-75}$.\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 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 $\boxed{-75}$.) |
math-013750 | Linear Algebra: Determinants — Cross-Validation | 7 | Solve and sanity-check: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&-3&4\\-2&-6&0\\5&6&-3\end{pmatrix}.$$
(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 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: $\\boxed{126}$.\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 y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{126}$.) |
math-013751 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Start by stating any domain restrictions: Compute the determinant of the matrix
$$A=\begin{pmatrix}0&3&-3\\6&-3&-5\\-2&3&-2\end{pmatrix}.$$
(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 ... | [
{
"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: $\\boxed{30}$.\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... | 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 $\boxed{30}$.) |
math-013752 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Provide a rigorous solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&2&3\\0&0&1\\-2&-6&1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods 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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-40}$.\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 $\boxed{-40}$.) |
math-013753 | Linear Algebra: Determinants — Cross-Validation | 7 | Prompt: Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&1&2\\-5&0&-2\\0&6&-4\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must 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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-152}$.\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. |
math-013754 | Linear Algebra: Determinants — Cross-Validation | 7 | Track quantifiers carefully: Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&-4&4\\1&-5&-6\\-4&5&3\end{pmatrix}.$$
(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 ... | [
{
"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: $\\boxed{-234}$.\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 $\boxed{-234}$.) |
math-013755 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Do not skip justification steps: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&-5&3\\-1&-2&-4\\-3&-5&5\end{pmatrix}.$$
(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 ... | [
{
"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: $\\boxed{-148}$.\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... | 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-013756 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Solve and include a self-check: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&2&6\\4&2&0\\-1&0&-6\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two 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: $\\boxed{48}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yi... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | 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-013757 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Find the exact value: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&4&-5\\3&6&-1\\4&5&6\end{pmatrix}.$$
(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 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": "The two methods are consistent and must coincide. Final answer: $\\boxed{203}$.\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 $\boxed{203}$.) |
math-013758 | Linear Algebra: Determinants — Row Operations | 7 | Prompt: Compute the determinant of the matrix
$$A=\begin{pmatrix}-4&-3&3\\3&-4&-2\\-1&-5&-5\end{pmatrix}.$$
(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.
When using ... | [
{
"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: $\\boxed{-148}$.\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-013759 | Linear Algebra: Determinants — Cross-Validation | 7 | Provide both a computational and a conceptual explanation: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&5&6\\0&1&-4\\6&4&-3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explai... | [
{
"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: $\\boxed{-143}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both meth... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-143}$.) |
math-013760 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Prompt: Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&5&-6\\6&0&-1\\-3&-2&2\end{pmatrix}.$$
(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 ro... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{39}$.\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 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 $\boxed{39}$.) |
math-013761 | Linear Algebra: Determinants — Cross-Validation | 7 | Problem: Compute the determinant of the matrix
$$A=\begin{pmatrix}-5&-6&-4\\-6&6&-3\\-4&-1&-5\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.
When usin... | [
{
"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: $\\boxed{153}$.\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 $\boxed{153}$.) |
math-013762 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Solve and sanity-check: Compute the determinant of the matrix
$$A=\begin{pmatrix}-3&6&-4\\2&-3&1\\5&3&-5\end{pmatrix}.$$
(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": "Both approaches agree after simplification. 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 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 $\boxed{-30}$.) |
math-013763 | Linear Algebra: Determinants — Cross-Validation | 7 | Problem: Compute the determinant of the matrix
$$A=\begin{pmatrix}-4&0&-3\\-4&-3&-3\\-1&4&0\end{pmatrix}.$$
(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 agree.
When using ... | [
{
"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: $\\boxed{9}$.\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-013764 | Matrix Theory: Determinant Properties | 7 | Work this out carefully: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&-5&0\\6&1&0\\-6&6&1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods 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: $\\boxed{31}$.\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 $\boxed{31}$.) |
math-013765 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Start by stating any domain restrictions: Compute the determinant of the matrix
$$A=\begin{pmatrix}5&5&3\\3&-4&-2\\-2&0&4\end{pmatrix}.$$
(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 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": "The two methods are consistent and must coincide. Final answer: $\\boxed{-144}$.\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. (Here the result is $\boxed{-144}$.) |
math-013766 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Show all reasoning: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&-5&2\\-2&-5&-5\\1&-4&-3\end{pmatrix}.$$
(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 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: $\\boxed{86}$.\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... | 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-013767 | Linear Algebra: Determinants — Row Operations | 7 | Problem: Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&-6&2\\-4&0&1\\2&-5&-2\end{pmatrix}.$$
(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 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": "The two methods are consistent and must coincide. 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 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... | 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 $\boxed{46}$.) |
math-013768 | Linear Algebra: Determinants — Row Operations | 7 | Track quantifiers carefully: Compute the determinant of the matrix
$$A=\begin{pmatrix}-3&-4&-2\\-1&-3&2\\1&-4&2\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-36}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-013769 | Matrix Theory: Determinant Properties | 7 | Solve and justify each step: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&0&-3\\-4&4&2\\2&6&-1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must ag... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{0}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yie... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-013770 | Linear Algebra: Determinants — Row Operations | 7 | Give a theorem-based solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&5&-4\\-5&2&-2\\-1&5&-6\end{pmatrix}.$$
(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 mu... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-36}$.\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 $\boxed{-36}$.) |
math-013771 | Linear Algebra: Determinants — Cross-Validation | 7 | Give a fully justified solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&-4&4\\-5&4&-2\\3&0&5\end{pmatrix}.$$
(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 mus... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-44}$.\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 $\boxed{-44}$.) |
math-013772 | Linear Algebra: Determinants — Row Operations | 7 | Provide a rigorous solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&2&2\\5&4&3\\4&6&2\end{pmatrix}.$$
(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": "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-013773 | Linear Algebra: Determinants — Cross-Validation | 7 | Provide a rigorous solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}5&4&-5\\1&-6&0\\-6&5&-4\end{pmatrix}.$$
(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 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: $\\boxed{291}$.\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-013774 | Linear Algebra: Determinants — Row Operations | 7 | Give a fully justified solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&-4&-5\\-4&5&1\\1&-2&-5\end{pmatrix}.$$
(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 m... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{15}$.\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 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 $\boxed{15}$.) |
math-013775 | Linear Algebra: Determinants — Row Operations | 7 | Indicate where a theorem is used: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&-6&6\\4&2&1\\5&3&1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods 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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{7}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | 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-013776 | Linear Algebra: Determinants — Cross-Validation | 7 | Show all reasoning: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&4&4\\3&-2&-3\\4&6&2\end{pmatrix}.$$
(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.
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": "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-013777 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Make each step logically reversible (or explain if not): Compute the determinant of the matrix
$$A=\begin{pmatrix}-5&4&0\\6&-1&6\\0&-3&6\end{pmatrix}.$$
(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... | [
{
"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: $\\boxed{-204}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same nu... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-204}$.) |
math-013778 | Linear Algebra: Determinants — Cross-Validation | 7 | Make each step logically reversible (or explain if not): Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&2&6\\-2&-3&6\\-6&1&1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explai... | [
{
"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: $\\boxed{-134}$.\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-013779 | Linear Algebra: Determinants — Row Operations | 7 | Warm-up: Compute the determinant of the matrix
$$A=\begin{pmatrix}-3&2&-5\\5&0&0\\-2&2&-2\end{pmatrix}.$$
(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 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": "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 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 $\boxed{-30}$.) |
math-013780 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Checkpoint: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&-4&-2\\2&-2&1\\6&-5&-6\end{pmatrix}.$$
(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 agree.
When usin... | [
{
"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: $\\boxed{-25}$.\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 $\boxed{-25}$.) |
math-013781 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Warm-up: Compute the determinant of the matrix
$$A=\begin{pmatrix}-2&-5&1\\0&-1&3\\5&-4&6\end{pmatrix}.$$
(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 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": "The two methods are consistent and must coincide. Final answer: $\\boxed{-82}$.\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-013782 | Linear Algebra: Determinants — Row Operations | 7 | Where appropriate, name the theorem you use: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&2&-6\\-1&0&2\\1&2&3\end{pmatrix}.$$
(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 ... | [
{
"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: $\\boxed{10}$.\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 $\boxed{10}$.) |
math-013783 | Linear Algebra: Determinants — Cross-Validation | 7 | Give a theorem-based solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}-2&-6&6\\-3&0&3\\5&3&-3\end{pmatrix}.$$
(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 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: $\\boxed{-72}$.\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 $\boxed{-72}$.) |
math-013784 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Give reasoning, not just computation: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&-6&2\\-2&2&3\\-5&-4&-4\end{pmatrix}.$$
(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 met... | [
{
"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: $\\boxed{170}$.\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. |
math-013785 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Explain why your operations are valid: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&2&2\\2&5&2\\6&6&-6\end{pmatrix}.$$
(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 method... | [
{
"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: $\\boxed{54}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both method... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{54}$.) |
math-013786 | Matrix Theory: Determinant Properties | 7 | Work this out carefully: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&-6&0\\-5&-2&-6\\-5&4&-3\end{pmatrix}.$$
(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 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: $\\boxed{-120}$.\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 $\boxed{-120}$.) |
math-013787 | Linear Algebra: Determinants — Cross-Validation | 7 | Prompt: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&2&-3\\-3&-5&-5\\5&-4&-3\end{pmatrix}.$$
(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 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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-199}$.\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... | 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 $\boxed{-199}$.) |
math-013788 | Linear Algebra: Determinants — Cross-Validation | 7 | Solve and sanity-check: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&-4&4\\-2&6&6\\2&-3&6\end{pmatrix}.$$
(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.
... | [
{
"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: $\\boxed{96}$.\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 $\boxed{96}$.) |
math-013789 | Matrix Theory: Determinant Properties | 7 | Solve and then verify: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&-5&-2\\3&1&4\\2&-1&-1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly 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: $\\boxed{-42}$.\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 $\boxed{-42}$.) |
math-013790 | Linear Algebra: Determinants — Row Operations | 7 | Track quantifiers carefully: Compute the determinant of the matrix
$$A=\begin{pmatrix}0&5&5\\4&-6&2\\1&0&1\end{pmatrix}.$$
(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 agre... | [
{
"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: $\\boxed{20}$.\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. |
math-013791 | Linear Algebra: Determinants — Cross-Validation | 7 | Use two approaches if possible: Compute the determinant of the matrix
$$A=\begin{pmatrix}-2&0&-5\\0&4&-4\\1&-1&2\end{pmatrix}.$$
(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 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: $\\boxed{12}$.\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 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. (Here the result is $\boxed{12}$.) |
math-013792 | Matrix Theory: Determinant Properties | 7 | Proceed methodically: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&0&-5\\1&2&4\\-2&5&-6\end{pmatrix}.$$
(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 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": "The two methods are consistent and must coincide. Final answer: $\\boxed{-141}$.\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 $\boxed{-141}$.) |
math-013793 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Compute the requested quantity: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&-2&5\\6&0&-6\\-2&-4&-4\end{pmatrix}.$$
(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 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: $\\boxed{-288}$.\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. (Here the result is $\boxed{-288}$.) |
math-013794 | Linear Algebra: Determinants — Row Operations | 7 | Derive the result step-by-step: Compute the determinant of the matrix
$$A=\begin{pmatrix}5&3&-4\\-6&-4&3\\6&6&3\end{pmatrix}.$$
(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 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": "The two methods are consistent and must coincide. 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 numbe... | [
{
"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 $\boxed{6}$.) |
math-013795 | Linear Algebra: Determinants — Cross-Validation | 7 | Provide a rigorous solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&-6&5\\-6&6&3\\-2&-5&-6\end{pmatrix}.$$
(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": "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: $\\boxed{420}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{420}$.) |
math-013796 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Work this out carefully: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&0&6\\2&-5&-4\\-3&-3&-5\end{pmatrix}.$$
(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 agre... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-100}$.\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-013797 | Linear Algebra: Determinants — Cross-Validation | 7 | Determine the requested value: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&-6&1\\4&0&1\\4&-3&4\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods 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": "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... | 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 $\boxed{66}$.) |
math-013798 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Do not skip justification steps: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&-2&0\\6&5&-2\\4&-3&5\end{pmatrix}.$$
(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 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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{190}$.\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-013799 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Give a fully justified solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}-4&-6&-6\\0&3&1\\0&3&0\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods mus... | [
{
"method_name": "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: $\\boxed{12}$.\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 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 $\boxed{12}$.) |
math-013800 | Linear Algebra: Determinants — Cofactor Expansion | 7 | Start by stating any domain restrictions: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&-4&-5\\0&-4&1\\-4&-1&-2\end{pmatrix}.$$
(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... | [
{
"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: $\\boxed{132}$.\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 $\boxed{132}$.) |
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