question 38 points 2

find $a^{-1}$.

\a = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2 \end{bmatrix}\

$\bigcirc$ $\frac{1}{18} \begin{bmatrix} -5 & 1 & 7 \\ 1 & 7 & -5 \\ 7 & -5 & 1 \end{bmatrix}$

$\bigcirc$ $\frac{1}{8} \begin{bmatrix} -5 & 1 & 7 \\ 1 & 7 & -5 \\ 7 & -5 & 1 \end{bmatrix}$

Question

question 38 points 2

find $a^{-1}$.

\a = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2 \end{bmatrix}\

$\bigcirc$ $\frac{1}{18} \begin{bmatrix} -5 & 1 & 7 \\ 1 & 7 & -5 \\ 7 & -5 & 1 \end{bmatrix}$

$\bigcirc$ $\frac{1}{8} \begin{bmatrix} -5 & 1 & 7 \\ 1 & 7 & -5 \\ 7 & -5 & 1 \end{bmatrix}$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Matrix Determinant", "Matrix Inverse Formula", "Cofactor Matrix", "Matrix Transpose" ], "new_concepts": [], "current_concepts": [ "Matrix Determinant", "Matrix Inverse Formula", "Cofactor Matrix", "Matrix Transpose" ] } ## Calculate the determinant of matrix A Using the **Matrix Determinant** knowledge point $$ \begin{aligned} \det(A) &= 1(3 \cdot 2 - 1 \cdot 1) - 2(2 \cdot 2 - 1 \cdot 3) + 3(2 \cdot 1 - 3 \cdot 3) \ &= 1(6 - 1) - 2(4 - 3) + 3(2 - 9) \ &= 5 - 2 - 21 \ &= -18 \end{aligned} $$ ## Find the cofactor matrix of A Using the **Cofactor Matrix** knowledge point $$ \begin{aligned} C_{11} &= +(3 \cdot 2 - 1 \cdot 1) = 5, \quad C_{12} = -(2 \cdot 2 - 1 \cdot 3) = -1, \quad C_{13} = +(2 \cdot 1 - 3 \cdot 3) = -7 \ C_{21} &= -(2 \cdot 2 - 3 \cdot 1) = -1, \quad C_{22} = +(1 \cdot 2 - 3 \cdot 3) = -7, \quad C_{23} = -(1 \cdot 1 - 3 \cdot 2) = 5 \ C_{31} &= +(2 \cdot 1 - 3 \cdot 3) = -7, \quad C_{32} = -(1 \cdot 1 - 2 \cdot 3) = 5, \quad C_{33} = +(1 \cdot 3 - 2 \cdot 2) = -1 \end{aligned} $$ $$ C = \begin{bmatrix} 5 & -1 & -7 \ -1 & -7 & 5 \ -7 & 5 & -1 \end{bmatrix} $$ ## Transpose the cofactor matrix to find the adjugate matrix Using the **Matrix Transpose** knowledge point $$ \text{adj}(A) = C^T = \begin{bmatrix} 5 & -1 & -7 \ -1 & -7 & 5 \ -7 & 5 & -1 \end{bmatrix} $$ ## Apply the matrix inverse formula Using the **Matrix Inverse Formula** knowledge point $$ \begin{aligned} A^{-1} &= \frac{1}{\det(A)} \text{adj}(A) \ &= -\frac{1}{18} \begin{bmatrix} 5 & -1 & -7 \ -1 & -7 & 5 \ -7 & 5 & -1 \end{bmatrix} \ &= \frac{1}{18} \begin{bmatrix} -5 & 1 & 7 \ 1 & 7 & -5 \ 7 & -5 & 1 \end{bmatrix} \end{aligned} $$

(A) $\frac{1}{18} \begin{bmatrix} -5 & 1 & 7 \ 1 & 7 & -5 \ 7 & -5 & 1 \end{bmatrix}$

(B) $\frac{1}{8} \begin{bmatrix} -5 & 1 & 7 \ 1 & 7 & -5 \ 7 & -5 & 1 \end{bmatrix}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Matrix Inverse Formula" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the determinant of matrix A

Find the cofactor matrix of A

Transpose the cofactor matrix to find the adjugate matrix

Apply the matrix inverse formula

Answer: