Question

find the inverse of the coefficient matrix \\(\

$$\begin{cases} -3x + 2y = 10 \\\\ 4x - 3y = 7 \\end{cases}$$

\\)

\\(\bigcirc \

$$\begin{bmatrix} -3 & -2 \\\\ -4 & -3 \\end{bmatrix}$$

\\)

\\(\bigcirc \

$$\begin{bmatrix} -3 & 2 \\\\ 4 & -3 \\end{bmatrix}$$

\\)

Explanation:

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<reasoning>

Identify the coefficient matrix

Using the Coefficient Matrix knowledge point
\[
A =

$$\begin{bmatrix} -3 & 2 \\ 4 & -3 \end{bmatrix}$$

\]

Calculate the determinant

Using the Matrix Determinant knowledge point
\[
\det(A) = (-3)(-3) - (2)(4) = 9 - 8 = 1
\]

Apply the inverse formula

Using the Matrix Inverse Formula knowledge point
\[
A^{-1} = \frac{1}{\det(A)}

$$\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$

= \frac{1}{1}

$$\begin{bmatrix} -3 & -2 \\ -4 & -3 \end{bmatrix}$$

=

$$\begin{bmatrix} -3 & -2 \\ -4 & -3 \end{bmatrix}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(

$$\begin{bmatrix} -3 & -2 \\ -4 & -3 \end{bmatrix}$$

\)</mcq-correct>
<mcq-option>(B) \(

$$\begin{bmatrix} -3 & 2 \\ 4 & -3 \end{bmatrix}$$

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

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Answer:

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<reasoning>

Identify the coefficient matrix

Using the Coefficient Matrix knowledge point
\[
A =

$$\begin{bmatrix} -3 & 2 \\ 4 & -3 \end{bmatrix}$$

\]

Calculate the determinant

Using the Matrix Determinant knowledge point
\[
\det(A) = (-3)(-3) - (2)(4) = 9 - 8 = 1
\]

Apply the inverse formula

Using the Matrix Inverse Formula knowledge point
\[
A^{-1} = \frac{1}{\det(A)}

$$\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$

= \frac{1}{1}

$$\begin{bmatrix} -3 & -2 \\ -4 & -3 \end{bmatrix}$$

=

$$\begin{bmatrix} -3 & -2 \\ -4 & -3 \end{bmatrix}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(

$$\begin{bmatrix} -3 & -2 \\ -4 & -3 \end{bmatrix}$$

\)</mcq-correct>
<mcq-option>(B) \(

$$\begin{bmatrix} -3 & 2 \\ 4 & -3 \end{bmatrix}$$

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

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