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} \

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

\\)

Explanation:

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<reasoning>

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
\[

$$\begin{aligned} A^{-1} &= \frac{1}{\det(A)} \text{adj}(A) \\ &= -\frac{1}{18} \begin{bmatrix} 5 & -1 & -7 \\ -1 & -7 & 5 \\ -7 & 5 & -1 \end{bmatrix} \\ &= \frac{1}{18} \begin{bmatrix} -5 & 1 & 7 \\ 1 & 7 & -5 \\ 7 & -5 & 1 \end{bmatrix} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(\frac{1}{18}

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

\)</mcq-correct>
<mcq-option>(B) \(\frac{1}{8}

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

\)</mcq-option>
</answer>

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Answer:

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<reasoning>

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
\[

$$\begin{aligned} A^{-1} &= \frac{1}{\det(A)} \text{adj}(A) \\ &= -\frac{1}{18} \begin{bmatrix} 5 & -1 & -7 \\ -1 & -7 & 5 \\ -7 & 5 & -1 \end{bmatrix} \\ &= \frac{1}{18} \begin{bmatrix} -5 & 1 & 7 \\ 1 & 7 & -5 \\ 7 & -5 & 1 \end{bmatrix} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(\frac{1}{18}

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

\)</mcq-correct>
<mcq-option>(B) \(\frac{1}{8}

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

\)</mcq-option>
</answer>

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