id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-011801 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Answer using clear logical steps: Solve the system and justify uniqueness:
$$\begin{cases}-22x+(25)y=503,\\9x+(20)y=429.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient ma... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}-22&25\\\\9&20\\end{pmatrix}$ has determinant $\\det(A)=-665\\ne 0$.... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(1,21)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-665\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(1,21)$.",
"robustness_analysis": "... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Remember: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(1,21)}$.) |
math-011802 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Explain each transformation: Compute the determinant of the matrix
$$A=\begin{pmatrix}5&2&6\\2&6&-5\\-3&3&-2\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two 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": "The two methods are consistent and must coincide. Final answer: $\\boxed{197}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both 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-011803 | Linear Algebra: Systems — Elimination | 6 | Checkpoint: Solve the system and justify uniqueness:
$$\begin{cases}-11x+(-17)y=-346,\\19x+(-14)y=164.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix being nonzero... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{(x,y)=(16,10)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=477\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(16,10)$.",
"robustness_an... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Remember: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(16,10)}$.) |
math-011804 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Indicate where a theorem is used: Solve the system and justify uniqueness:
$$\begin{cases}-11x+(-14)y=-471,\\20x+(10)y=470.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{(x,y)=(11,25)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=170\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(11,25)$.",
"robustness_an... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Remember: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011805 | Linear Algebra: Determinants — Row Operations | 6 | Complete the analysis: Compute the determinant of the matrix
$$A=\begin{pmatrix}5&-3&-3\\-5&4&6\\-1&-3&2\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{61}$.\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... | 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 $\boxed{61}$.) |
math-011806 | Linear Algebra: Determinant and Uniqueness | 6 | Try to avoid pattern-matching; explain why: Solve the system and justify uniqueness:
$$\begin{cases}20x+(-2)y=172,\\12x+(-10)y=332.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coe... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(6,-26)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-176\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(6,-26)$.",
"robustness_analysis":... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Core principle: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(6,-26)}$.) |
math-011807 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Derive the result step-by-step: Compute the determinant of the matrix
$$A=\begin{pmatrix}-2&3&0\\3&-5&3\\4&2&-3\end{pmatrix}.$$
(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 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: $\\boxed{45}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both method... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-011808 | Linear Algebra: Determinants — Cross-Validation | 6 | Answer with a short justification: Compute the determinant of the matrix
$$A=\begin{pmatrix}-5&1&3\\-3&-1&6\\1&5&1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two 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: $\\boxed{122}$.\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 $\boxed{122}$.) |
math-011809 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Solve (and briefly cross-validate): Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&0&1\\-6&1&-3\\-2&-1&-1\end{pmatrix}.$$
(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 metho... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{32}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both 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. |
math-011810 | Linear Algebra: Determinants — Cross-Validation | 6 | Solve and justify each step: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&-5&-2\\-5&1&-6\\2&6&0\end{pmatrix}.$$
(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 must a... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{340}$.\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-011811 | Linear Algebra: Systems — Cramer's Rule | 6 | Indicate where a theorem is used: Solve the system and justify uniqueness:
$$\begin{cases}12x+(-13)y=-208,\\-2x+(-2)y=68.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient m... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(-26,-8)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-50\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-26,-8)$.",
"robustness_analysis": "Generality not... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Key idea: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(-26,-8)}$.) |
math-011812 | Linear Algebra: Systems — Cramer's Rule | 6 | Give reasoning, not just computation: Solve the system and justify uniqueness:
$$\begin{cases}-19x+(-6)y=-84,\\12x+(-23)y=696.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coeffici... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(12,-24)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=509\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(12,-24)$.",
"robustness_analysis"... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Remember: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011813 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Do not skip justification steps: Solve the system and justify uniqueness:
$$\begin{cases}11x+(8)y=82,\\-20x+(6)y=-334.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matr... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(14,-9)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=226\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(14,-9)$.",
"robustness_analysis": ... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Core principle: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011814 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Solve with verification: Compute the determinant of the matrix
$$A=\begin{pmatrix}0&-2&1\\-2&-4&0\\-3&3&0\end{pmatrix}.$$
(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 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: $\\boxed{-18}$.\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-011815 | Linear Algebra: Systems — Cramer's Rule | 6 | Question: Solve the system and justify uniqueness:
$$\begin{cases}22x+(-8)y=-688,\\-2x+(20)y=236.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix being nonzero.
In... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}22&-8\\\\-2&20\\end{pmatrix}$ has determinant $\\det(A)=424\\ne 0$."... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(-28,9)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=424\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-28,9)$.",
"robustness_analysis": "Sensitivity anal... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Core principle: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(-28,9)}$.) |
math-011816 | Matrix Theory: Determinant Properties | 6 | Challenge: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&-3&4\\3&-6&-3\\-5&-5&6\end{pmatrix}.$$
(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": "Both approaches agree after simplification. 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 number."... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-120}$.) |
math-011817 | Linear Algebra: Systems — Cramer's Rule | 6 | Be explicit about assumptions: Solve the system and justify uniqueness:
$$\begin{cases}-17x+(-18)y=552,\\11x+(2)y=-280.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient mat... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(-24,-8)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=164\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-24,-8)$.",
"robustness_analysis": "Robustness not... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Remember: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(-24,-8)}$.) |
math-011818 | Matrix Theory: Determinant Properties | 6 | Find the exact value: Compute the determinant of the matrix
$$A=\begin{pmatrix}-3&-4&-3\\-6&-2&-4\\3&-6&-3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two 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: $\\boxed{48}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same 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 $\boxed{48}$.) |
math-011819 | Linear Algebra: Determinant and Uniqueness | 6 | Indicate where a theorem is used: Solve the system and justify uniqueness:
$$\begin{cases}-15x+(12)y=195,\\12x+(17)y=243.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient m... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}-15&12\\\\12&17\\end{pmatrix}$ has determinant $\\det(A)=-399\\ne 0$... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(-1,15)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-399\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-1,15)$.",
"robustness_analysis": "If the problem were p... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Core principle: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011820 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Show all reasoning: Compute the determinant of the matrix
$$A=\begin{pmatrix}0&4&3\\-5&-5&0\\6&-3&-4\end{pmatrix}.$$
(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 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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{55}$.\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 $\boxed{55}$.) |
math-011821 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Solve and include a self-check: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&0&6\\6&-1&-1\\-6&6&3\end{pmatrix}.$$
(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 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: $\\boxed{177}$.\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 $\boxed{177}$.) |
math-011822 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | State any required conditions first: Solve the system and justify uniqueness:
$$\begin{cases}24x+(-10)y=-144,\\1x+(-3)y=-6.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(-6,0)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-62\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-6,0)$.",
"robustness_analysis": "Sensitivity analysis: El... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Core principle: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(-6,0)}$.) |
math-011823 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Provide a rigorous solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&6&2\\3&6&6\\-1&-4&5\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agr... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{186}$.\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 $\boxed{186}$.) |
math-011824 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Write the solution set clearly: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&0&4\\-2&5&-5\\0&6&5\end{pmatrix}.$$
(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": "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: $\\boxed{7}$.\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 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 $\boxed{7}$.) |
math-011825 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Solve and sanity-check: Compute the determinant of the matrix
$$A=\begin{pmatrix}5&-5&3\\3&-1&-5\\4&0&-2\end{pmatrix}.$$
(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.... | [
{
"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: $\\boxed{92}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both method... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-011826 | Linear Algebra: Determinants — Row Operations | 6 | Solve and then verify: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&0&4\\-1&-2&-4\\-5&-3&6\end{pmatrix}.$$
(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": "Consistency verification shows both paths yield the identical boxed result. 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 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. (Here the result is $\boxed{-4}$.) |
math-011827 | Linear Algebra: Systems — Elimination | 6 | Give a fully justified solution: Solve the system and justify uniqueness:
$$\begin{cases}-25x+(19)y=710,\\6x+(5)y=-27.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matr... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}-25&19\\\\6&5\\end{pmatrix}$ has determinant $\\det(A)=-239\\ne 0$."... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(-17,15)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-239\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-17,15)$.",
"robustness_analysis": "Robustness note: El... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Remember: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(-17,15)}$.) |
math-011828 | Matrix Theory: Determinant Properties | 6 | State any required conditions first: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&4&1\\4&-1&-5\\-4&3&0\end{pmatrix}.$$
(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 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": "The two methods are consistent and must coincide. Final answer: $\\boxed{73}$.\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-011829 | Linear Algebra: Systems — Cramer's Rule | 6 | Compute the requested quantity: Solve the system and justify uniqueness:
$$\begin{cases}20x+(-6)y=-52,\\-25x+(20)y=-35.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient mat... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}20&-6\\\\-25&20\\end{pmatrix}$ has determinant $\\det(A)=250\\ne 0$.... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(-5,-8)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=250\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-5,-8)$.",
"robustness_analysis": ... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Core principle: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011830 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Explain each transformation: Solve the system and justify uniqueness:
$$\begin{cases}10x+(24)y=452,\\5x+(-17)y=-441.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{(x,y)=(-10,23)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-290\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-10,23)$.",
"robustness... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Takeaway: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011831 | Linear Algebra: Determinant and Uniqueness | 6 | Solve (and briefly cross-validate): Solve the system and justify uniqueness:
$$\begin{cases}-13x+(-17)y=-661,\\10x+(-2)y=222.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficie... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(26,19)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=196\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(26,19)$.",
"robustness_analysis": "Generality note: Elimi... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Core principle: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(26,19)}$.) |
math-011832 | Linear Algebra: Systems — Elimination | 6 | Explain why your operations are valid: Solve the system and justify uniqueness:
$$\begin{cases}-1x+(-13)y=303,\\-4x+(-5)y=84.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficie... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{(x,y)=(9,-24)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-47\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(9,-24)$.",
"robustness_an... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Takeaway: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011833 | Linear Algebra: Determinant and Uniqueness | 6 | Keep the final answer in boxed form: Solve the system and justify uniqueness:
$$\begin{cases}1x+(-18)y=227,\\13x+(14)y=-521.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficien... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{(x,y)=(-25,-14)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=248\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-25,-14)$.",
"robustnes... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Key idea: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011834 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Carefully track domains: Compute the determinant of the matrix
$$A=\begin{pmatrix}-5&6&-4\\0&1&5\\-5&-2&-1\end{pmatrix}.$$
(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 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: $\\boxed{-215}$.\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 $\boxed{-215}$.) |
math-011835 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Problem: Solve the system and justify uniqueness:
$$\begin{cases}-18x+(-23)y=109,\\-19x+(13)y=-295.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix being nonzero.
... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(8,-11)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-671\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(8,-11)$.",
"robustness_analysis":... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Takeaway: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(8,-11)}$.) |
math-011836 | Matrix Theory: Determinant Properties | 6 | Answer using clear logical steps: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&-2&0\\0&2&-6\\2&-3&-5\end{pmatrix}.$$
(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 m... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-88}$.\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-011837 | Linear Algebra: Systems — Elimination | 6 | Do not skip justification steps: Solve the system and justify uniqueness:
$$\begin{cases}3x+(-13)y=193,\\13x+(-3)y=143.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient mat... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(8,-13)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=160\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(8,-13)$.",
"robustness_analysis": ... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Takeaway: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(8,-13)}$.) |
math-011838 | Linear Algebra: Systems — Cramer's Rule | 6 | Do not skip justification steps: Solve the system and justify uniqueness:
$$\begin{cases}11x+(23)y=-573,\\-9x+(8)y=-148.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient ma... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(-4,-23)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=295\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-4,-23)$.",
"robustness_analysis"... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Key idea: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(-4,-23)}$.) |
math-011839 | Linear Algebra: Systems — Cramer's Rule | 6 | Solve and justify each step: Solve the system and justify uniqueness:
$$\begin{cases}-10x+(7)y=62,\\15x+(-20)y=-340.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}-10&7\\\\15&-20\\end{pmatrix}$ has determinant $\\det(A)=95\\ne 0$."... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(12,26)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=95\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(12,26)$.",
"robustness_analysis": "Robustness note: Elimin... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Key idea: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011840 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Explain each transformation: Solve the system and justify uniqueness:
$$\begin{cases}-17x+(14)y=-465,\\8x+(9)y=-15.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix ... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{(x,y)=(15,-15)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-265\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(15,-15)$.",
"robustness... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Takeaway: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011841 | Matrix Theory: Determinant Properties | 6 | Solve and then verify: Compute the determinant of the matrix
$$A=\begin{pmatrix}5&4&2\\-5&-2&-4\\-3&-1&2\end{pmatrix}.$$
(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 must agree.... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{46}$.\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 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 $\boxed{46}$.) |
math-011842 | Linear Algebra: Systems — Elimination | 6 | Try to avoid pattern-matching; explain why: Solve the system and justify uniqueness:
$$\begin{cases}1x+(-1)y=42,\\-11x+(22)y=-781.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coef... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(13,-29)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=11\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(13,-29)$.",
"robustness_analysis": "Robustness note: Elim... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Core principle: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(13,-29)}$.) |
math-011843 | Linear Algebra: Systems — Cramer's Rule | 6 | Work carefully and justify each inference: Solve the system and justify uniqueness:
$$\begin{cases}-12x+(-10)y=-64,\\-24x+(-13)y=-142.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the ... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(7,-2)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-84\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(7,-2)$.",
"robustness_analysis": "Generality note: E... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Key idea: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011844 | Linear Algebra: Systems — Elimination | 6 | Do not skip justification steps: Solve the system and justify uniqueness:
$$\begin{cases}10x+(8)y=244,\\20x+(8)y=304.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matri... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}10&8\\\\20&8\\end{pmatrix}$ has determinant $\\det(A)=-80\\ne 0$.",
... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(6,23)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-80\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(6,23)$.",
"robustness_analysis": "I... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Takeaway: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011845 | Linear Algebra: Determinant and Uniqueness | 6 | Explain why your operations are valid: Solve the system and justify uniqueness:
$$\begin{cases}-3x+(-8)y=-22,\\-3x+(20)y=-134.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coeffici... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}-3&-8\\\\-3&20\\end{pmatrix}$ has determinant $\\det(A)=-84\\ne 0$."... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{(x,y)=(18,-4)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-84\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(18,-4)$.",
"robustness_an... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Takeaway: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(18,-4)}$.) |
math-011846 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Solve and sanity-check: Solve the system and justify uniqueness:
$$\begin{cases}21x+(21)y=-357,\\14x+(19)y=-383.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix bei... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(12,-29)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=105\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(12,-29)$.",
"robustness_analysis": "Generality note: Eli... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Core principle: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011847 | Linear Algebra: Determinants — Row Operations | 6 | Solve (and briefly cross-validate): Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&-3&4\\2&5&-4\\-6&0&5\end{pmatrix}.$$
(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... | [
{
"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: $\\boxed{53}$.\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 $\boxed{53}$.) |
math-011848 | Linear Algebra: Determinant and Uniqueness | 6 | Proceed methodically: Solve the system and justify uniqueness:
$$\begin{cases}19x+(-3)y=109,\\-13x+(10)y=-162.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix being... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(4,-11)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=151\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(4,-11)$.",
"robustness_analysis": "Sensitivity anal... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Takeaway: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011849 | Linear Algebra: Determinants — Cross-Validation | 6 | Compute the requested quantity: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&0&-2\\-5&1&1\\-4&1&-6\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods mus... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-19}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same num... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-19}$.) |
math-011850 | Linear Algebra: Determinant and Uniqueness | 6 | Find the exact value: Solve the system and justify uniqueness:
$$\begin{cases}14x+(22)y=-688,\\-18x+(-13)y=640.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix bein... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(-24,-16)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=214\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-24,-16)$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Takeaway: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(-24,-16)}$.) |
math-011851 | Linear Algebra: Systems — Cramer's Rule | 6 | Give a fully justified solution: Solve the system and justify uniqueness:
$$\begin{cases}19x+(16)y=-195,\\-7x+(15)y=-158.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient m... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{(x,y)=(-1,-11)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=397\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-1,-11)$.",
"robustness_... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Key idea: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(-1,-11)}$.) |
math-011852 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Provide both a computational and a conceptual explanation: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&0&-5\\2&-5&-6\\6&3&4\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly expla... | [
{
"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: $\\boxed{-192}$.\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 $\boxed{-192}$.) |
math-011853 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | State any required conditions first: Solve the system and justify uniqueness:
$$\begin{cases}14x+(-23)y=694,\\3x+(22)y=-336.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficien... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}14&-23\\\\3&22\\end{pmatrix}$ has determinant $\\det(A)=377\\ne 0$."... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{(x,y)=(20,-18)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=377\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(20,-18)$.",
"robustness_... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Remember: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(20,-18)}$.) |
math-011854 | Linear Algebra: Systems — Elimination | 6 | Work carefully and justify each inference: Solve the system and justify uniqueness:
$$\begin{cases}24x+(3)y=456,\\12x+(10)y=24.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coeffic... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}24&3\\\\12&10\\end{pmatrix}$ has determinant $\\det(A)=204\\ne 0$.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(22,-24)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=204\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(22,-24)$.",
"robustness_analysis"... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Remember: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011855 | Linear Algebra: Determinant and Uniqueness | 6 | Checkpoint: Solve the system and justify uniqueness:
$$\begin{cases}15x+(3)y=-12,\\-13x+(-1)y=-28.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix being nonzero.
I... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}15&3\\\\-13&-1\\end{pmatrix}$ has determinant $\\det(A)=24\\ne 0$.",... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{(x,y)=(4,-24)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=24\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(4,-24)$.",
"robustness_ana... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Key idea: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(4,-24)}$.) |
math-011856 | Linear Algebra: Systems — Cramer's Rule | 6 | Carefully track domains: Solve the system and justify uniqueness:
$$\begin{cases}-11x+(-11)y=231,\\20x+(-19)y=-303.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix ... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}-11&-11\\\\20&-19\\end{pmatrix}$ has determinant $\\det(A)=429\\ne 0... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{(x,y)=(-18,-3)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=429\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-18,-3)$.",
"robustness_... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Remember: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(-18,-3)}$.) |
math-011857 | Linear Algebra: Determinants — Cross-Validation | 6 | Keep the final answer in boxed form: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&4&-2\\5&3&3\\-2&-1&3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two 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": "The two methods are consistent and must coincide. Final answer: $\\boxed{-62}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same num... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-62}$.) |
math-011858 | Linear Algebra: Systems — Cramer's Rule | 6 | Solve and include a self-check: Solve the system and justify uniqueness:
$$\begin{cases}2x+(22)y=-136,\\8x+(21)y=-142.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matr... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}2&22\\\\8&21\\end{pmatrix}$ has determinant $\\det(A)=-134\\ne 0$.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(-2,-6)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-134\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-2,-6)$.",
"robustness_analysis": "Generality note... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Remember: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011859 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | State any required conditions first: Solve the system and justify uniqueness:
$$\begin{cases}14x+(8)y=196,\\15x+(16)y=262.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient ... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{(x,y)=(10,7)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=104\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(10,7)$.",
"robustness_anal... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Core principle: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(10,7)}$.) |
math-011860 | Linear Algebra: Determinants — Cross-Validation | 6 | Use two approaches if possible: Compute the determinant of the matrix
$$A=\begin{pmatrix}-4&1&-6\\2&-2&4\\6&-4&-6\end{pmatrix}.$$
(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 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: $\\boxed{-100}$.\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 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-011861 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Give an answer and a quick verification: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&2&0\\1&4&-2\\-5&-6&5\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two 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": "Both approaches agree after simplification. Final answer: $\\boxed{26}$.\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 $\boxed{26}$.) |
math-011862 | Linear Algebra: Determinants — Row Operations | 6 | Show all reasoning: Compute the determinant of the matrix
$$A=\begin{pmatrix}-3&0&-5\\4&4&-4\\-6&-3&2\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must 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": "Both approaches agree after simplification. Final answer: $\\boxed{-48}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | 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 $\boxed{-48}$.) |
math-011863 | Linear Algebra: Determinants — Row Operations | 6 | Provide both a computational and a conceptual explanation: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&-2&-2\\-3&-2&6\\6&-3&-2\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly 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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-124}$.\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 $\boxed{-124}$.) |
math-011864 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Use two approaches if possible: Solve the system and justify uniqueness:
$$\begin{cases}19x+(-7)y=-80,\\-20x+(-23)y=-98.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient ma... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(-2,6)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-577\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-2,6)$.",
"robustness_analysis": "Robustness note: Elimin... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Remember: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(-2,6)}$.) |
math-011865 | Linear Algebra: Determinant and Uniqueness | 6 | Give a fully justified solution: Solve the system and justify uniqueness:
$$\begin{cases}11x+(-4)y=-186,\\13x+(-7)y=-163.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient m... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(-26,-25)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-25\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-26,-25)$.",
"robustness_analysi... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Takeaway: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011866 | Linear Algebra: Determinant and Uniqueness | 6 | Carefully track domains: Solve the system and justify uniqueness:
$$\begin{cases}-13x+(-15)y=211,\\10x+(-1)y=101.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix be... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{(x,y)=(8,-21)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=163\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(8,-21)$.",
"robustness_an... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Core principle: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(8,-21)}$.) |
math-011867 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Track quantifiers carefully: Solve the system and justify uniqueness:
$$\begin{cases}2x+(-10)y=256,\\-17x+(5)y=64.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix b... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(-12,-28)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-160\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-12,-28)$.",
"robustness_analysis": "Generality ... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Remember: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(-12,-28)}$.) |
math-011868 | Linear Algebra: Determinant and Uniqueness | 6 | Complete the analysis: Solve the system and justify uniqueness:
$$\begin{cases}25x+(3)y=472,\\-24x+(23)y=-1126.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix bein... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}25&3\\\\-24&23\\end{pmatrix}$ has determinant $\\det(A)=647\\ne 0$."... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(22,-26)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=647\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(22,-26)$.",
"robustness_analysis": "If the problem were ... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Core principle: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(22,-26)}$.) |
math-011869 | Linear Algebra: Determinants — Row Operations | 6 | Where appropriate, name the theorem you use: Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&0&4\\-2&4&1\\5&4&-4\end{pmatrix}.$$
(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... | [
{
"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: $\\boxed{8}$.\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. (Here the result is $\boxed{8}$.) |
math-011870 | Linear Algebra: Systems — Elimination | 6 | Solve with verification: Solve the system and justify uniqueness:
$$\begin{cases}19x+(-24)y=456,\\7x+(8)y=-152.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix bein... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}19&-24\\\\7&8\\end{pmatrix}$ has determinant $\\det(A)=320\\ne 0$.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(0,-19)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=320\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(0,-19)$.",
"robustness_analysis": ... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Key idea: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(0,-19)}$.) |
math-011871 | Linear Algebra: Determinants — Cross-Validation | 6 | Try to avoid pattern-matching; explain why: Compute the determinant of the matrix
$$A=\begin{pmatrix}5&-1&-5\\4&3&-1\\4&-6&2\end{pmatrix}.$$
(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... | [
{
"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: $\\boxed{192}$.\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-011872 | Matrix Theory: Determinant Properties | 6 | Task: Compute the determinant of the matrix
$$A=\begin{pmatrix}5&0&-3\\0&-3&4\\1&6&-4\end{pmatrix}.$$
(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 agree.
When using row re... | [
{
"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: $\\boxed{-69}$.\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-011873 | Linear Algebra: Determinants — Cross-Validation | 6 | Task: Compute the determinant of the matrix
$$A=\begin{pmatrix}0&-1&-4\\-2&2&-5\\0&-1&-6\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods 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: $\\boxed{4}$.\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... | 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-011874 | Linear Algebra: Systems — Elimination | 6 | Keep the final answer in boxed form: Solve the system and justify uniqueness:
$$\begin{cases}-20x+(24)y=-888,\\-4x+(17)y=-385.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coeffici... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(24,-17)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-244\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(24,-17)$.",
"robustness_analysis": "Generality note: El... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Remember: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011875 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Keep the final answer in boxed form: Solve the system and justify uniqueness:
$$\begin{cases}-25x+(21)y=-267,\\-12x+(16)y=-288.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coeffic... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(-12,-27)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-148\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-12,-27)$.",
"robustness_analysis": "Robustness ... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Takeaway: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011876 | Linear Algebra: Determinant and Uniqueness | 6 | Solve and include a self-check: Solve the system and justify uniqueness:
$$\begin{cases}14x+(-19)y=415,\\-14x+(2)y=44.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matr... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(-7,-27)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-238\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-7,-27)$.",
"robustness_analysis... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Takeaway: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(-7,-27)}$.) |
math-011877 | Linear Algebra: Systems — Cramer's Rule | 6 | Derive the result step-by-step: Solve the system and justify uniqueness:
$$\begin{cases}3x+(9)y=240,\\18x+(-5)y=24.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix ... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}3&9\\\\18&-5\\end{pmatrix}$ has determinant $\\det(A)=-177\\ne 0$.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(8,24)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-177\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(8,24)$.",
"robustness_analysis": "Generality note: Elimin... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Key idea: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(8,24)}$.) |
math-011878 | Linear Algebra: Determinant and Uniqueness | 6 | Be explicit about assumptions: Solve the system and justify uniqueness:
$$\begin{cases}12x+(12)y=132,\\-21x+(23)y=-847.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient mat... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}12&12\\\\-21&23\\end{pmatrix}$ has determinant $\\det(A)=528\\ne 0$.... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(25,-14)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=528\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(25,-14)$.",
"robustness_analysis"... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Core principle: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(25,-14)}$.) |
math-011879 | Linear Algebra: Determinants — Cross-Validation | 6 | Provide a rigorous solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}-3&5&0\\2&-4&-2\\-4&0&3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two 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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{46}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both 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 $\boxed{46}$.) |
math-011880 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Be explicit about assumptions: Solve the system and justify uniqueness:
$$\begin{cases}11x+(16)y=-71,\\11x+(7)y=-62.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{(x,y)=(-5,-1)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-99\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-5,-1)$.",
"robustness_an... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Core principle: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011881 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Warm-up: Solve the system and justify uniqueness:
$$\begin{cases}12x+(-22)y=462,\\12x+(6)y=42.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix being nonzero.
Inclu... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(11,-15)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=336\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(11,-15)$.",
"robustness_analysis"... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Key idea: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011882 | Matrix Theory: Determinant Properties | 6 | Prompt: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&6&-2\\-6&0&5\\-3&3&-3\end{pmatrix}.$$
(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 agree.
When using row... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-222}$.\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 $\boxed{-222}$.) |
math-011883 | Linear Algebra: Determinant and Uniqueness | 6 | Provide a rigorous solution: Solve the system and justify uniqueness:
$$\begin{cases}-25x+(13)y=-482,\\7x+(-25)y=-100.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matr... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(25,11)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=534\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(25,11)$.",
"robustness_analysis": ... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Key idea: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(25,11)}$.) |
math-011884 | Linear Algebra: Determinant and Uniqueness | 6 | Prompt: Solve the system and justify uniqueness:
$$\begin{cases}18x+(-13)y=-773,\\8x+(9)y=85.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix being nonzero.
Includ... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}18&-13\\\\8&9\\end{pmatrix}$ has determinant $\\det(A)=266\\ne 0$.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(-22,29)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=266\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-22,29)$.",
"robustness_analysis": "If the problem were ... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Remember: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(-22,29)}$.) |
math-011885 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Give an answer and a quick verification: Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&-2&1\\-2&-2&2\\1&-1&-4\end{pmatrix}.$$
(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": "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: $\\boxed{-44}$.\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 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-011886 | Linear Algebra: Systems — Cramer's Rule | 6 | Show all reasoning: Solve the system and justify uniqueness:
$$\begin{cases}4x+(3)y=26,\\8x+(-5)y=-58.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix being nonzero... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}4&3\\\\8&-5\\end{pmatrix}$ has determinant $\\det(A)=-44\\ne 0$.",
... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(-1,10)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-44\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-1,10)$.",
"robustness_analysis": "Sensitivity analysis: ... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Takeaway: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(-1,10)}$.) |
math-011887 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Task: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&-3&-3\\3&-6&-4\\-5&2&-4\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.
When using 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": "Both approaches agree after simplification. Final answer: $\\boxed{72}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",
... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | 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-011888 | Matrix Theory: Determinant Properties | 6 | Give a fully justified solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}-2&4&3\\-1&-5&-4\\-1&0&1\end{pmatrix}.$$
(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 m... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{15}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both method... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-011889 | Linear Algebra: Systems — Cramer's Rule | 6 | Explain what is being counted/optimized: Solve the system and justify uniqueness:
$$\begin{cases}-17x+(9)y=476,\\-2x+(19)y=361.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coeffic... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}-17&9\\\\-2&19\\end{pmatrix}$ has determinant $\\det(A)=-305\\ne 0$.... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(-19,17)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-305\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-19,17)$.",
"robustness_analysis": "Sensitivity analysi... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Remember: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(-19,17)}$.) |
math-011890 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Question: Solve the system and justify uniqueness:
$$\begin{cases}-11x+(-3)y=7,\\11x+(19)y=249.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix being nonzero.
Incl... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}-11&-3\\\\11&19\\end{pmatrix}$ has determinant $\\det(A)=-176\\ne 0$... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(-5,16)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-176\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-5,16)$.",
"robustness_analysis": "Robustness note: Elim... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Remember: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(-5,16)}$.) |
math-011891 | Matrix Theory: Determinant Properties | 6 | Explain what is being counted/optimized: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&4&6\\-4&-5&-4\\6&-3&6\end{pmatrix}.$$
(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 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: $\\boxed{210}$.\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 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 $\boxed{210}$.) |
math-011892 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Answer with a short justification: Compute the determinant of the matrix
$$A=\begin{pmatrix}0&4&4\\2&4&3\\-1&5&1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two 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: $\\boxed{36}$.\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 $\boxed{36}$.) |
math-011893 | Linear Algebra: Systems — Elimination | 6 | Provide a rigorous solution: Solve the system and justify uniqueness:
$$\begin{cases}-20x+(8)y=216,\\6x+(12)y=-252.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix ... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{(x,y)=(-16,-13)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-288\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-16,-13)$.",
"robustne... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Key idea: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(-16,-13)}$.) |
math-011894 | Linear Algebra: Systems — Elimination | 6 | Provide a rigorous solution: Solve the system and justify uniqueness:
$$\begin{cases}-5x+(22)y=218,\\-18x+(-12)y=-36.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matri... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(-4,9)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=456\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-4,9)$.",
"robustness_analysis": "Sensitivity analys... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Core principle: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. |
math-011895 | Linear Algebra: Determinants — Row Operations | 6 | Track quantifiers carefully: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&-4&-2\\0&1&-6\\3&-6&5\end{pmatrix}.$$
(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 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: $\\boxed{47}$.\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 $\boxed{47}$.) |
math-011896 | Linear Algebra: Systems — Cramer's Rule | 6 | Compute the requested quantity: Solve the system and justify uniqueness:
$$\begin{cases}-18x+(-14)y=380,\\-11x+(13)y=-242.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient ... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(-4,-22)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-388\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-4,-22)$.",
"robustness_analysis": "If the proble... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Remember: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(-4,-22)}$.) |
math-011897 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Work carefully and justify each inference: Solve the system and justify uniqueness:
$$\begin{cases}-22x+(-5)y=-553,\\-12x+(-24)y=60.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the co... | [
{
"method_name": "Cramer's Rule",
"approach": "Use determinants: for $A\\mathbf{x}=\\mathbf{b}$ with $\\det(A)\\ne 0$, the unique solution is given by Cramer's formulas.",
"steps": [
"Step 1: Coefficient matrix $A=\\begin{pmatrix}-22&-5\\\\-12&-24\\end{pmatrix}$ has determinant $\\det(A)=468\\ne 0... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(29,-17)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=468\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(29,-17)$.",
"robustness_analysis": "Generality note: Eli... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Key idea: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(29,-17)}$.) |
math-011898 | Linear Algebra: Systems — Cramer's Rule | 6 | Challenge: Solve the system and justify uniqueness:
$$\begin{cases}21x+(4)y=58,\\-21x+(-25)y=299.\end{cases}$$
(a) Solve by elimination.
(b) Solve using Cramer's rule.
(c) Explain why the solution must be unique.
In part (c), explicitly relate uniqueness to the determinant of the coefficient matrix being nonzero.
In... | [
{
"method_name": "Elimination",
"approach": "Eliminate one variable by forming a linear combination, then back-substitute.",
"steps": [
"Step 1: Multiply equations if needed so that adding/subtracting cancels one variable.",
"Step 2: Solve the resulting single-variable equation.",
"Ste... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{(x,y)=(6,-17)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-441\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(6,-17)$.",
"robustness_a... | [
{
"error_description": "Tried to eliminate without matching coefficients first.",
"why_plausible": "The idea 'add equations to eliminate' can be misapplied.",
"why_wrong": "Variables cancel only when coefficients are equal/opposite; otherwise elimination fails and produces incorrect equations.",
"wh... | Key idea: A $2\times2$ system has a unique solution exactly when the determinant of its coefficient matrix is nonzero; elimination and determinant methods must then agree. (Here the result is $\boxed{(x,y)=(6,-17)}$.) |
math-011899 | Linear Algebra: Determinants — Row Operations | 6 | Question: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&-2&-6\\1&5&1\\0&0&5\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.
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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{160}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both metho... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{160}$.) |
math-011900 | Matrix Theory: Determinant Properties | 6 | Be explicit about assumptions: Compute the determinant of the matrix
$$A=\begin{pmatrix}-5&5&-4\\-5&1&3\\-4&5&-1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two 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: $\\boxed{79}$.\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. |
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