id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-011701 | Linear Algebra: Systems — Cramer's Rule | 6 | Indicate where a theorem is used: Solve the system and justify uniqueness:
$$\begin{cases}9x+(-15)y=135,\\10x+(20)y=150.\end{cases}$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(15,0)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=330\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(15,0)$.",
"robustness_analysis": "If the problem wer... | [
{
"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 $\boxed{(x,y)=(15,0)}$.) |
math-011702 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Show all reasoning: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&2&-6\\-1&-1&-5\\5&5&6\end{pmatrix}.$$
(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.
W... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-57}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-011703 | Linear Algebra: Systems — Cramer's Rule | 6 | Try to avoid pattern-matching; explain why: Solve the system and justify uniqueness:
$$\begin{cases}20x+(18)y=-128,\\-1x+(-15)y=232.\end{cases}$$
(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": "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: $\\boxed{(x,y)=(8,-16)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-282\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(8,-16)$.",
"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 $\boxed{(x,y)=(8,-16)}$.) |
math-011704 | Linear Algebra: Systems — Cramer's Rule | 6 | Explain each transformation: Solve the system and justify uniqueness:
$$\begin{cases}4x+(17)y=-456,\\-6x+(11)y=-192.\end{cases}$$
(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 $A=\\begin{pmatrix}4&17\\\\-6&11\\end{pmatrix}$ has determinant $\\det(A)=146\\ne 0$.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(-12,-24)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=146\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-12,-24)$.",
"robustness_analysis": "Robustness 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. (Here the result is $\boxed{(x,y)=(-12,-24)}$.) |
math-011705 | Linear Algebra: Determinants — Row Operations | 6 | Compute the requested quantity: Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&-3&2\\5&0&-5\\1&1&4\end{pmatrix}.$$
(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 methods must... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{55}$.\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-011706 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Derive the result step-by-step: Compute the determinant of the matrix
$$A=\begin{pmatrix}5&0&5\\6&0&2\\-3&-2&3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must ... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-40}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | 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-011707 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Task: Compute the determinant of the matrix
$$A=\begin{pmatrix}-2&3&5\\6&3&-5\\-1&4&1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods 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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{86}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yi... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Takeaway: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{86}$.) |
math-011708 | Linear Algebra: Systems — Cramer's Rule | 6 | Give a theorem-based solution: Solve the system and justify uniqueness:
$$\begin{cases}9x+(-10)y=136,\\-12x+(13)y=-175.\end{cases}$$
(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 $A=\\begin{pmatrix}9&-10\\\\-12&13\\end{pmatrix}$ has determinant $\\det(A)=-3\\ne 0$."... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(-6,-19)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-3\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-6,-19)$.",
"robustness_analysis": "If the problem ... | [
{
"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-011709 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Track units/moduli carefully: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&1&-6\\4&2&-2\\-1&-3&-1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{18}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both method... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{18}$.) |
math-011710 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Work carefully and justify each inference: Solve the system and justify uniqueness:
$$\begin{cases}-8x+(6)y=-148,\\-5x+(16)y=-166.\end{cases}$$
(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": "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 $A=\\begin{pmatrix}-8&6\\\\-5&16\\end{pmatrix}$ has determinant $\\det(A)=-98\\ne 0$.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(14,-6)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-98\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(14,-6)$.",
"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... | 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-011711 | Matrix Theory: Determinant Properties | 6 | Do not skip justification steps: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&3&3\\6&-1&-1\\-4&1&-4\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods 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": "Cross-check: both derivations land on the same invariant quantity. 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 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-011712 | Linear Algebra: Determinants — Cross-Validation | 6 | Be explicit about assumptions: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&5&-2\\5&-2&5\\3&-2&-6\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods mus... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{211}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{211}$.) |
math-011713 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Show all reasoning: Solve the system and justify uniqueness:
$$\begin{cases}-5x+(-10)y=-5,\\-17x+(-2)y=-17.\end{cases}$$
(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 no... | [
{
"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: $\\boxed{(x,y)=(1,0)}$.\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 $(x,y)=(1,0)$.",
"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... | 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-011714 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Write the solution set clearly: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&-6&-6\\1&-1&2\\5&-3&-5\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods mu... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-80}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | 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-011715 | Linear Algebra: Determinants — Row Operations | 6 | Give a theorem-based solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}-2&-1&3\\-3&2&4\\0&-5&-3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods mus... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{26}$.\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 yi... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Takeaway: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-011716 | Linear Algebra: Determinants — Row Operations | 6 | Try to avoid pattern-matching; explain why: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&-3&-2\\6&0&5\\-3&-5&-2\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the t... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. 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 method... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{44}$.) |
math-011717 | Matrix Theory: Determinant Properties | 6 | Give a theorem-based solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}-5&1&-5\\0&3&-3\\-4&-1&-1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods mu... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-18}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-18}$.) |
math-011718 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Provide both a computational and a conceptual explanation: Solve the system and justify uniqueness:
$$\begin{cases}19x+(25)y=598,\\-21x+(-17)y=-342.\end{cases}$$
(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 deter... | [
{
"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: $\\boxed{(x,y)=(-8,30)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=202\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-8,30)$.",
"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 $\boxed{(x,y)=(-8,30)}$.) |
math-011719 | Matrix Theory: Determinant Properties | 6 | State any required conditions first: Compute the determinant of the matrix
$$A=\begin{pmatrix}2&-2&6\\-3&-2&-3\\-3&-4&-3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two met... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{24}$.\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 $\boxed{24}$.) |
math-011720 | Linear Algebra: Systems — Cramer's Rule | 6 | Checkpoint: Solve the system and justify uniqueness:
$$\begin{cases}-7x+(13)y=-275,\\5x+(-24)y=373.\end{cases}$$
(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 $A=\\begin{pmatrix}-7&13\\\\5&-24\\end{pmatrix}$ has determinant $\\det(A)=103\\ne 0$."... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(17,-12)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=103\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(17,-12)$.",
"robustness_analysis": "Sensitivity 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 $\boxed{(x,y)=(17,-12)}$.) |
math-011721 | Linear Algebra: Determinant and Uniqueness | 6 | Indicate where a theorem is used: Solve the system and justify uniqueness:
$$\begin{cases}-11x+(-5)y=-21,\\7x+(-7)y=217.\end{cases}$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{(x,y)=(11,-20)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=112\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(11,-20)$.",
"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 $\boxed{(x,y)=(11,-20)}$.) |
math-011722 | Linear Algebra: Systems — Cramer's Rule | 6 | Start by stating any domain restrictions: Solve the system and justify uniqueness:
$$\begin{cases}-4x+(-21)y=-558,\\9x+(24)y=558.\end{cases}$$
(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 coeff... | [
{
"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: $\\boxed{(x,y)=(-18,30)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=93\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-18,30)$.",
"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 $\boxed{(x,y)=(-18,30)}$.) |
math-011723 | Linear Algebra: Determinant and Uniqueness | 6 | Prompt: Solve the system and justify uniqueness:
$$\begin{cases}7x+(22)y=228,\\-7x+(-1)y=150.\end{cases}$$
(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 $A=\\begin{pmatrix}7&22\\\\-7&-1\\end{pmatrix}$ has determinant $\\det(A)=147\\ne 0$.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(-24,18)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=147\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-24,18)$.",
"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 $\boxed{(x,y)=(-24,18)}$.) |
math-011724 | Linear Algebra: Systems — Cramer's Rule | 6 | Prompt: Solve the system and justify uniqueness:
$$\begin{cases}-4x+(16)y=-124,\\-20x+(-22)y=298.\end{cases}$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(-5,-9)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=408\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-5,-9)$.",
"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... | 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-011725 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Challenge: Solve the system and justify uniqueness:
$$\begin{cases}-21x+(21)y=189,\\-20x+(-17)y=69.\end{cases}$$
(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: $\\boxed{(x,y)=(-6,3)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=777\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-6,3)$.",
"robustness_analysis": "S... | [
{
"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 $\boxed{(x,y)=(-6,3)}$.) |
math-011726 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Track quantifiers carefully: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&-5&1\\0&-4&-5\\-3&3&-6\end{pmatrix}.$$
(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 ... | [
{
"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: $\\boxed{-48}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both metho... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-48}$.) |
math-011727 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Answer using clear logical steps: Solve the system and justify uniqueness:
$$\begin{cases}19x+(15)y=403,\\13x+(-6)y=-17.\end{cases}$$
(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: $\\boxed{(x,y)=(7,18)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-309\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(7,18)$.",
"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 $\boxed{(x,y)=(7,18)}$.) |
math-011728 | Linear Algebra: Determinants — Row Operations | 6 | Do not skip justification steps: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&3&6\\-2&5&-4\\-2&5&5\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods mus... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{324}$.\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-011729 | Linear Algebra: Systems — Elimination | 6 | Indicate where a theorem is used: Solve the system and justify uniqueness:
$$\begin{cases}19x+(6)y=-241,\\-18x+(5)y=239.\end{cases}$$
(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: $\\boxed{(x,y)=(-13,1)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=203\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-13,1)$.",
"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. |
math-011730 | Linear Algebra: Determinants — Cross-Validation | 6 | Work this out carefully: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&5&5\\-6&3&-3\\6&4&-6\end{pmatrix}.$$
(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 must agree.... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-486}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same nu... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-486}$.) |
math-011731 | Linear Algebra: Systems — Elimination | 6 | Determine the requested value: Solve the system and justify uniqueness:
$$\begin{cases}4x+(13)y=58,\\-19x+(23)y=233.\end{cases}$$
(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: $\\boxed{(x,y)=(-5,6)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=339\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-5,6)$.",
"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... | 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 $\boxed{(x,y)=(-5,6)}$.) |
math-011732 | Linear Algebra: Systems — Cramer's Rule | 6 | Work this out carefully: Solve the system and justify uniqueness:
$$\begin{cases}-24x+(3)y=-363,\\19x+(-25)y=-233.\end{cases}$$
(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": "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 $A=\\begin{pmatrix}-24&3\\\\19&-25\\end{pmatrix}$ has determinant $\\det(A)=543\\ne 0$.... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(18,23)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=543\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(18,23)$.",
"robustness_analysis": "Robustness 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 $\boxed{(x,y)=(18,23)}$.) |
math-011733 | Matrix Theory: Determinant Properties | 6 | Task: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&6&-3\\-1&-1&-1\\1&3&2\end{pmatrix}.$$
(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.
When using row r... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{18}$.\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 yi... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Takeaway: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{18}$.) |
math-011734 | Matrix Theory: Determinant Properties | 6 | Explain what is being counted/optimized: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&-4&-3\\1&-2&5\\-1&-5&-5\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two ... | [
{
"method_name": "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: $\\boxed{56}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yi... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{56}$.) |
math-011735 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Explain why your operations are valid: Solve the system and justify uniqueness:
$$\begin{cases}22x+(-7)y=570,\\-10x+(19)y=-354.\end{cases}$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{(x,y)=(24,-6)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=348\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(24,-6)$.",
"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 $\boxed{(x,y)=(24,-6)}$.) |
math-011736 | Linear Algebra: Determinant and Uniqueness | 6 | Complete the analysis: Solve the system and justify uniqueness:
$$\begin{cases}-7x+(21)y=-567,\\-10x+(-2)y=-74.\end{cases}$$
(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 $A=\\begin{pmatrix}-7&21\\\\-10&-2\\end{pmatrix}$ has determinant $\\det(A)=224\\ne 0$.... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(12,-23)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=224\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(12,-23)$.",
"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... | 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 $\boxed{(x,y)=(12,-23)}$.) |
math-011737 | Linear Algebra: Determinant and Uniqueness | 6 | Give an answer and a quick verification: Solve the system and justify uniqueness:
$$\begin{cases}-23x+(-16)y=49,\\-10x+(19)y=800.\end{cases}$$
(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 coeff... | [
{
"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 $A=\\begin{pmatrix}-23&-16\\\\-10&19\\end{pmatrix}$ has determinant $\\det(A)=-597\\ne ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(-23,30)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-597\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-23,30)$.",
"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 $\boxed{(x,y)=(-23,30)}$.) |
math-011738 | Linear Algebra: Systems — Cramer's Rule | 6 | Carefully track domains: Solve the system and justify uniqueness:
$$\begin{cases}8x+(-14)y=636,\\-24x+(-14)y=-228.\end{cases}$$
(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": "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 $A=\\begin{pmatrix}8&-14\\\\-24&-14\\end{pmatrix}$ has determinant $\\det(A)=-448\\ne 0... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(27,-30)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-448\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(27,-30)$.",
"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-011739 | Linear Algebra: Determinants — Row Operations | 6 | Question: Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&0&4\\-4&3&-6\\2&4&-6\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.
When using r... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-124}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must ... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-011740 | Linear Algebra: Determinants — Row Operations | 6 | Explain what is being counted/optimized: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&-5&-4\\-3&2&3\\-3&-4&-1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two ... | [
{
"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: $\\boxed{-2}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yi... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Takeaway: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-2}$.) |
math-011741 | Matrix Theory: Determinant Properties | 6 | State any required conditions first: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&2&-3\\4&3&5\\1&-3&5\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods ... | [
{
"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: $\\boxed{135}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both metho... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Takeaway: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{135}$.) |
math-011742 | Linear Algebra: Determinants — Row Operations | 6 | Write the solution set clearly: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&6&-2\\-4&-2&2\\-6&-2&-6\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods m... | [
{
"method_name": "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: $\\boxed{-192}$.\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 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 $\boxed{-192}$.) |
math-011743 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Solve and sanity-check: Solve the system and justify uniqueness:
$$\begin{cases}10x+(-5)y=315,\\-17x+(22)y=-765.\end{cases}$$
(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": "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 $A=\\begin{pmatrix}10&-5\\\\-17&22\\end{pmatrix}$ has determinant $\\det(A)=135\\ne 0$.... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(23,-17)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=135\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(23,-17)$.",
"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-011744 | Linear Algebra: Determinants — Cross-Validation | 6 | Keep the final answer in boxed form: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&-2&-3\\0&1&2\\3&-4&-2\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two method... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{33}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same numb... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{33}$.) |
math-011745 | Linear Algebra: Determinant and Uniqueness | 6 | Write the solution set clearly: Solve the system and justify uniqueness:
$$\begin{cases}18x+(4)y=308,\\21x+(5)y=364.\end{cases}$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(14,14)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=6\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(14,14)$.",
"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... | 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 $\boxed{(x,y)=(14,14)}$.) |
math-011746 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Complete the analysis: Solve the system and justify uniqueness:
$$\begin{cases}8x+(3)y=-47,\\-17x+(-5)y=104.\end{cases}$$
(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 n... | [
{
"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 $A=\\begin{pmatrix}8&3\\\\-17&-5\\end{pmatrix}$ has determinant $\\det(A)=11\\ne 0$.",
... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(-7,3)}$.\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 $(x,y)=(-7,3)$.",
"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... | 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 $\boxed{(x,y)=(-7,3)}$.) |
math-011747 | Linear Algebra: Systems — Cramer's Rule | 6 | Answer using clear logical steps: Solve the system and justify uniqueness:
$$\begin{cases}-4x+(-22)y=-156,\\-6x+(11)y=-58.\end{cases}$$
(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 ... | [
{
"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 $A=\\begin{pmatrix}-4&-22\\\\-6&11\\end{pmatrix}$ has determinant $\\det(A)=-176\\ne 0$... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(17,4)}$.\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 $(x,y)=(17,4)$.",
"robustness_analysis": "If the problem were per... | [
{
"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-011748 | Linear Algebra: Determinant and Uniqueness | 6 | Exercise: Solve the system and justify uniqueness:
$$\begin{cases}10x+(1)y=136,\\19x+(5)y=184.\end{cases}$$
(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: $\\boxed{(x,y)=(16,-24)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=31\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(16,-24)$.",
"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... | 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 $\boxed{(x,y)=(16,-24)}$.) |
math-011749 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Solve and sanity-check: Solve the system and justify uniqueness:
$$\begin{cases}9x+(6)y=-324,\\23x+(5)y=-704.\end{cases}$$
(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": "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 $A=\\begin{pmatrix}9&6\\\\23&5\\end{pmatrix}$ has determinant $\\det(A)=-93\\ne 0$.",
... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(-28,-12)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-93\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-28,-12)$.",
"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... | 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 $\boxed{(x,y)=(-28,-12)}$.) |
math-011750 | Linear Algebra: Determinants — Cross-Validation | 6 | Checkpoint: Compute the determinant of the matrix
$$A=\begin{pmatrix}-3&-3&3\\2&5&-3\\0&-4&1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.
When using... | [
{
"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: $\\boxed{3}$.\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... | 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-011751 | Linear Algebra: Determinants — Cross-Validation | 6 | Determine the requested value: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&-2&2\\1&-5&-2\\1&-2&-2\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods mu... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{0}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yie... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Takeaway: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{0}$.) |
math-011752 | Linear Algebra: Systems — Cramer's Rule | 6 | Task: Solve the system and justify uniqueness:
$$\begin{cases}-1x+(-14)y=87,\\19x+(16)y=97.\end{cases}$$
(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.
Include ... | [
{
"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 $A=\\begin{pmatrix}-1&-14\\\\19&16\\end{pmatrix}$ has determinant $\\det(A)=250\\ne 0$.... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(11,-7)}$.\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 $(x,y)=(11,-7)$.",
"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 $\boxed{(x,y)=(11,-7)}$.) |
math-011753 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Give a theorem-based solution: Solve the system and justify uniqueness:
$$\begin{cases}-8x+(-25)y=-498,\\3x+(25)y=593.\end{cases}$$
(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 $A=\\begin{pmatrix}-8&-25\\\\3&25\\end{pmatrix}$ has determinant $\\det(A)=-125\\ne 0$.... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(-19,26)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-125\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-19,26)$.",
"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... | 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 $\boxed{(x,y)=(-19,26)}$.) |
math-011754 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Determine the requested value: Solve the system and justify uniqueness:
$$\begin{cases}-12x+(19)y=127,\\-8x+(-18)y=-682.\end{cases}$$
(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: $\\boxed{(x,y)=(29,25)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=368\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(29,25)$.",
"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 $\boxed{(x,y)=(29,25)}$.) |
math-011755 | Linear Algebra: Systems — Cramer's Rule | 6 | Work this out carefully: Solve the system and justify uniqueness:
$$\begin{cases}13x+(-10)y=202,\\-7x+(15)y=-128.\end{cases}$$
(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": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(14,-2)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=125\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(14,-2)$.",
"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 $\boxed{(x,y)=(14,-2)}$.) |
math-011756 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Challenge: Solve the system and justify uniqueness:
$$\begin{cases}9x+(24)y=-150,\\-15x+(7)y=62.\end{cases}$$
(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 being nonzero.
Inc... | [
{
"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: $\\boxed{(x,y)=(-6,-4)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=423\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-6,-4)$.",
"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 $\boxed{(x,y)=(-6,-4)}$.) |
math-011757 | Linear Algebra: Determinant and Uniqueness | 6 | Explain why your operations are valid: Solve the system and justify uniqueness:
$$\begin{cases}23x+(-16)y=204,\\24x+(-13)y=102.\end{cases}$$
(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": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(-12,-30)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=85\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-12,-30)$.",
"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... | 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-011758 | Linear Algebra: Determinant and Uniqueness | 6 | Provide a rigorous solution: Solve the system and justify uniqueness:
$$\begin{cases}22x+(11)y=297,\\-17x+(-12)y=-128.\end{cases}$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{(x,y)=(28,-29)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-77\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(28,-29)$.",
"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... | 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-011759 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Where appropriate, name the theorem you use: Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&4&-6\\4&1&0\\3&1&4\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two ... | [
{
"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: $\\boxed{-94}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-011760 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Give an answer and a quick verification: Solve the system and justify uniqueness:
$$\begin{cases}22x+(8)y=166,\\9x+(-23)y=-405.\end{cases}$$
(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 $A=\\begin{pmatrix}22&8\\\\9&-23\\end{pmatrix}$ has determinant $\\det(A)=-578\\ne 0$."... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(1,18)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-578\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(1,18)$.",
"robustness_analysis": "Robustness 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... | 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 $\boxed{(x,y)=(1,18)}$.) |
math-011761 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Find the exact value: Solve the system and justify uniqueness:
$$\begin{cases}4x+(-8)y=-52,\\-14x+(-2)y=242.\end{cases}$$
(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 matrix being n... | [
{
"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 $A=\\begin{pmatrix}4&-8\\\\-14&-2\\end{pmatrix}$ has determinant $\\det(A)=-120\\ne 0$.... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(-17,-2)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-120\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-17,-2)$.",
"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. |
math-011762 | Linear Algebra: Systems — Cramer's Rule | 6 | Where appropriate, name the theorem you use: Solve the system and justify uniqueness:
$$\begin{cases}-10x+(-15)y=585,\\-7x+(-17)y=572.\end{cases}$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(-21,-25)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=65\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-21,-25)$.",
"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 $\boxed{(x,y)=(-21,-25)}$.) |
math-011763 | Matrix Theory: Determinant Properties | 6 | Solve (and briefly cross-validate): Compute the determinant of the matrix
$$A=\begin{pmatrix}-2&1&3\\3&6&-4\\-4&-3&-5\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two method... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{160}$.\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... | 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-011764 | Linear Algebra: Systems — Elimination | 6 | State any required conditions first: Solve the system and justify uniqueness:
$$\begin{cases}-18x+(-2)y=-188,\\14x+(20)y=-278.\end{cases}$$
(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 $A=\\begin{pmatrix}-18&-2\\\\14&20\\end{pmatrix}$ has determinant $\\det(A)=-332\\ne 0$... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(13,-23)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-332\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(13,-23)$.",
"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... | 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 $\boxed{(x,y)=(13,-23)}$.) |
math-011765 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Do not skip justification steps: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&5&-2\\-3&6&-1\\-1&1&6\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods mu... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{237}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same num... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{237}$.) |
math-011766 | Linear Algebra: Determinant and Uniqueness | 6 | State any required conditions first: Solve the system and justify uniqueness:
$$\begin{cases}17x+(-9)y=-399,\\2x+(-4)y=-94.\end{cases}$$
(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: $\\boxed{(x,y)=(-15,16)}$.\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 $(x,y)=(-15,16)$.",
"robustness_analysis": "Sensitivity 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... | 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 $\boxed{(x,y)=(-15,16)}$.) |
math-011767 | Linear Algebra: Systems — Elimination | 6 | Explain what is being counted/optimized: Solve the system and justify uniqueness:
$$\begin{cases}-23x+(3)y=60,\\-21x+(-20)y=123.\end{cases}$$
(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 coeffi... | [
{
"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: $\\boxed{(x,y)=(-3,-3)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=523\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-3,-3)$.",
"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 $\boxed{(x,y)=(-3,-3)}$.) |
math-011768 | Linear Algebra: Determinants — Cross-Validation | 6 | State any required conditions first: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&-3&-2\\-1&0&4\\2&-2&1\end{pmatrix}.$$
(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 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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-39}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must 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 $\boxed{-39}$.) |
math-011769 | Linear Algebra: Systems — Cramer's Rule | 6 | Provide a rigorous solution: Solve the system and justify uniqueness:
$$\begin{cases}-24x+(-13)y=-51,\\-16x+(15)y=321.\end{cases}$$
(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 $A=\\begin{pmatrix}-24&-13\\\\-16&15\\end{pmatrix}$ has determinant $\\det(A)=-568\\ne ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(-6,15)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-568\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-6,15)$.",
"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... | 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 $\boxed{(x,y)=(-6,15)}$.) |
math-011770 | Linear Algebra: Systems — Cramer's Rule | 6 | Explain each transformation: Solve the system and justify uniqueness:
$$\begin{cases}-10x+(-17)y=-281,\\7x+(5)y=107.\end{cases}$$
(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 $A=\\begin{pmatrix}-10&-17\\\\7&5\\end{pmatrix}$ has determinant $\\det(A)=69\\ne 0$.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(6,13)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=69\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(6,13)$.",
"robustness_analysis": "Generality note: Eliminat... | [
{
"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 $\boxed{(x,y)=(6,13)}$.) |
math-011771 | Linear Algebra: Determinants — Row Operations | 6 | Work carefully and justify each inference: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&1&0\\4&-1&6\\-5&-5&-2\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two ... | [
{
"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: $\\boxed{10}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same numb... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-011772 | Matrix Theory: Determinant Properties | 6 | Carefully track domains: Compute the determinant of the matrix
$$A=\begin{pmatrix}-4&1&-5\\-2&-1&-5\\-1&-4&0\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must ag... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{50}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same numb... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Key idea: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{50}$.) |
math-011773 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Indicate where a theorem is used: Solve the system and justify uniqueness:
$$\begin{cases}16x+(10)y=556,\\-15x+(-12)y=-600.\end{cases}$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{(x,y)=(16,30)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-42\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(16,30)$.",
"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 $\boxed{(x,y)=(16,30)}$.) |
math-011774 | Linear Algebra: Systems — Elimination | 6 | Keep the final answer in boxed form: Solve the system and justify uniqueness:
$$\begin{cases}3x+(8)y=-279,\\14x+(3)y=-478.\end{cases}$$
(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: $\\boxed{(x,y)=(-29,-24)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-103\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-29,-24)$.",
"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... | 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-011775 | Matrix Theory: Determinant Properties | 6 | Answer using clear logical steps: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&6&5\\-5&-1&3\\-5&-5&5\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods 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: $\\boxed{220}$.\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-011776 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Write the solution set clearly: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&4&2\\0&6&-6\\-5&-1&2\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must... | [
{
"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: $\\boxed{204}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same number.",... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. |
math-011777 | Linear Algebra: Determinant and Uniqueness | 6 | Be explicit about assumptions: Solve the system and justify uniqueness:
$$\begin{cases}-18x+(-9)y=-684,\\-24x+(23)y=-212.\end{cases}$$
(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: $\\boxed{(x,y)=(28,20)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-630\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(28,20)$.",
"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 $\boxed{(x,y)=(28,20)}$.) |
math-011778 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Track units/moduli carefully: Solve the system and justify uniqueness:
$$\begin{cases}-4x+(-10)y=-294,\\3x+(-23)y=-359.\end{cases}$$
(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 $A=\\begin{pmatrix}-4&-10\\\\3&-23\\end{pmatrix}$ has determinant $\\det(A)=122\\ne 0$.... | {
"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 $\\det(A)=122\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(26,19)$.",
"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... | 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-011779 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Proceed methodically: Compute the determinant of the matrix
$$A=\begin{pmatrix}-1&-3&4\\3&-1&-2\\4&0&0\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.
... | [
{
"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: $\\boxed{40}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same numb... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Takeaway: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{40}$.) |
math-011780 | Linear Algebra: Determinants — Cross-Validation | 6 | Explain what is being counted/optimized: Compute the determinant of the matrix
$$A=\begin{pmatrix}-4&-1&5\\-3&5&3\\-4&-3&-3\end{pmatrix}.$$
(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 ... | [
{
"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: $\\boxed{190}$.\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-011781 | Linear Algebra: Determinant and Uniqueness | 6 | Complete the analysis: Solve the system and justify uniqueness:
$$\begin{cases}-12x+(-11)y=358,\\4x+(-3)y=-26.\end{cases}$$
(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": "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 $A=\\begin{pmatrix}-12&-11\\\\4&-3\\end{pmatrix}$ has determinant $\\det(A)=80\\ne 0$."... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{(x,y)=(-17,-14)}$.\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 $(x,y)=(-17,-14)$.",
"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 $\boxed{(x,y)=(-17,-14)}$.) |
math-011782 | Linear Algebra: Systems — Cramer's Rule | 6 | Solve with verification: Solve the system and justify uniqueness:
$$\begin{cases}19x+(24)y=20,\\11x+(5)y=145.\end{cases}$$
(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": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(20,-15)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-169\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(20,-15)$.",
"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... | 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-011783 | Linear Algebra: Determinant and Uniqueness | 6 | Give reasoning, not just computation: Solve the system and justify uniqueness:
$$\begin{cases}13x+(-16)y=369,\\1x+(-6)y=57.\end{cases}$$
(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": "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 $A=\\begin{pmatrix}13&-16\\\\1&-6\\end{pmatrix}$ has determinant $\\det(A)=-62\\ne 0$."... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(21,-6)}$.\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 $(x,y)=(21,-6)$.",
"robustness_analysis": "If the problem were pe... | [
{
"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-011784 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Provide both a computational and a conceptual explanation: Solve the system and justify uniqueness:
$$\begin{cases}7x+(-9)y=224,\\-9x+(-21)y=-60.\end{cases}$$
(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 determin... | [
{
"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 $A=\\begin{pmatrix}7&-9\\\\-9&-21\\end{pmatrix}$ has determinant $\\det(A)=-228\\ne 0$.... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(23,-7)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-228\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(23,-7)$.",
"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... | 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 $\boxed{(x,y)=(23,-7)}$.) |
math-011785 | Linear Algebra: Systems — Cramer's Rule | 6 | Show all reasoning: Solve the system and justify uniqueness:
$$\begin{cases}19x+(20)y=-218,\\18x+(-13)y=688.\end{cases}$$
(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 n... | [
{
"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: $\\boxed{(x,y)=(18,-28)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-607\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(18,-28)$.",
"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 $\boxed{(x,y)=(18,-28)}$.) |
math-011786 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Give a theorem-based solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}4&3&-3\\3&-5&5\\5&0&3\end{pmatrix}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-87}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-87}$.) |
math-011787 | Linear Algebra: Systems — Elimination | 6 | Provide both a computational and a conceptual explanation: Solve the system and justify uniqueness:
$$\begin{cases}-25x+(18)y=-276,\\22x+(2)y=564.\end{cases}$$
(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 determi... | [
{
"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: $\\boxed{(x,y)=(24,18)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-446\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(24,18)$.",
"robustness_analysis": "If the problem ... | [
{
"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-011788 | Linear Algebra: Determinants — Cross-Validation | 6 | Provide a rigorous solution: Compute the determinant of the matrix
$$A=\begin{pmatrix}6&1&-3\\-5&-1&-3\\-3&2&-5\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must... | [
{
"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: $\\boxed{89}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same 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-011789 | Linear Algebra: Determinants — Cofactor Expansion | 6 | Answer using clear logical steps: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&3&0\\2&-1&0\\1&3&-3\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods 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: $\\boxed{27}$.\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-011790 | Matrix Theory: Determinant Properties | 6 | Show all reasoning: Compute the determinant of the matrix
$$A=\begin{pmatrix}-2&-5&2\\-1&3&-5\\4&-6&-4\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods 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: $\\boxed{192}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both metho... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Remember: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{192}$.) |
math-011791 | Matrix Theory: Determinant Properties | 6 | Try to avoid pattern-matching; explain why: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&-3&0\\6&6&2\\-3&-1&-2\end{pmatrix}.$$
(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... | [
{
"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: $\\boxed{-28}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same num... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Takeaway: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{-28}$.) |
math-011792 | Linear Algebra: Determinants — Cross-Validation | 6 | Compute the requested quantity: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&2&-2\\-4&1&6\\2&-6&1\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{25}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same numb... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{25}$.) |
math-011793 | Matrix Theory: Determinant Properties | 6 | Answer with a short justification: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&4&-2\\0&6&0\\1&-4&2\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods 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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{24}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both 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 $\boxed{24}$.) |
math-011794 | Linear Algebra: Systems — Elimination | 6 | Try to avoid pattern-matching; explain why: Solve the system and justify uniqueness:
$$\begin{cases}1x+(-4)y=-121,\\-21x+(3)y=678.\end{cases}$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{(x,y)=(-29,23)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=-81\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-29,23)$.",
"robustness_analysis": "If the problem... | [
{
"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 $\boxed{(x,y)=(-29,23)}$.) |
math-011795 | Linear Algebra: Determinants — Cross-Validation | 6 | Write the solution set clearly: Compute the determinant of the matrix
$$A=\begin{pmatrix}-4&6&-6\\6&-1&-6\\1&-6&2\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods mu... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{254}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must y... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | Core principle: Determinants can be computed by expansion or row operations; knowing exactly how each row operation affects $\det$ is essential for correct row-reduction computations. (Here the result is $\boxed{254}$.) |
math-011796 | Algebra: Linear Systems — Consistency/Uniqueness | 6 | Answer using clear logical steps: Solve the system and justify uniqueness:
$$\begin{cases}24x+(20)y=-856,\\-7x+(-4)y=213.\end{cases}$$
(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 $A=\\begin{pmatrix}24&20\\\\-7&-4\\end{pmatrix}$ has determinant $\\det(A)=44\\ne 0$.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(-19,-20)}$.\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 $(x,y)=(-19,-20)$.",
"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 $\boxed{(x,y)=(-19,-20)}$.) |
math-011797 | Linear Algebra: Determinant and Uniqueness | 6 | Track units/moduli carefully: Solve the system and justify uniqueness:
$$\begin{cases}10x+(-17)y=-245,\\-5x+(22)y=190.\end{cases}$$
(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 $A=\\begin{pmatrix}10&-17\\\\-5&22\\end{pmatrix}$ has determinant $\\det(A)=135\\ne 0$.... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{(x,y)=(-16,5)}$.\nElimination and Cramer's rule solve the same linear system. Since $\\det(A)=135\\ne 0$, the solution is unique, so both methods must agree on $(x,y)=(-16,5)$.",
"robustness_analysis": "Robustness 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... | 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-011798 | Matrix Theory: Determinant Properties | 6 | Proceed methodically: Compute the determinant of the matrix
$$A=\begin{pmatrix}3&-4&-2\\3&5&-2\\5&5&0\end{pmatrix}.$$
(a) Compute $\det(A)$ by cofactor expansion.
(b) Compute $\det(A)$ by row reduction, carefully tracking how row operations affect the determinant.
(c) Briefly explain why the two methods must agree.
W... | [
{
"method_name": "Cofactor Expansion",
"approach": "Expand along a convenient row/column to reduce to $2\\times2$ determinants.",
"steps": [
"Step 1: Choose a row/column (often one with zeros, if present).",
"Step 2: Compute the signed minors (2×2 determinants) and sum with the checkerboard ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{90}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both method... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | 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-011799 | Linear Algebra: Determinants — Row Operations | 6 | Explain what is being counted/optimized: Compute the determinant of the matrix
$$A=\begin{pmatrix}1&-3&2\\2&3&4\\0&-3&-3\end{pmatrix}.$$
(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 met... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-27}$.\nCofactor expansion and row-reduction compute the same multilinear alternating function (the determinant). Because the determinant is uniquely characterized by these properties, both methods must yield the same num... | [
{
"error_description": "Forgot the checkerboard signs in cofactor expansion.",
"why_plausible": "The minors are the most visible part and signs are easy to omit.",
"why_wrong": "Cofactors alternate in sign; missing signs changes the determinant.",
"which_method_catches_it": "Row-reduction method pro... | 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 $\boxed{-27}$.) |
math-011800 | Linear Algebra: Determinants — Cross-Validation | 6 | Solve (and briefly cross-validate): Compute the determinant of the matrix
$$A=\begin{pmatrix}-6&4&5\\6&-5&-3\\5&-2&-4\end{pmatrix}.$$
(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 method... | [
{
"method_name": "Row Reduction with Determinant Tracking",
"approach": "Use row operations to reach an upper-triangular matrix; the determinant is the product of diagonal entries times the tracked sign/scale factors.",
"steps": [
"Step 1: Apply row operations to create zeros below the diagonal.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{17}$.\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 $\boxed{17}$.) |
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