`{:[(-6,5),(-7,6)]^(-1)=:}`

Options

A. `{:[(-6,5),(-7,6)]:}`
B. `{:[(6,-5),(-7,6)]:}`
C. `{:[(6,5),(7,6)]:}`
D. `{:[(6,-5),(7,-6)]:}`

Answer

To find the inverse of the matrix \( A = \begin{pmatrix} -6 & 5 \\ -7 & 6 \end{pmatrix} \), we will follow the steps outlined below:

### Step 1: Calculate the Determinant of Matrix A
The determinant of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by the formula:
\[ \text{det}(A) = ad - bc \]
For our matrix \( A \):
- \( a = -6 \)
- \( b = 5 \)
- \( c = -7 \)
- \( d = 6 \)

Now, substituting these values into the determinant formula:
\[ \text{det}(A) = (-6)(6) - (5)(-7) = -36 + 35 = -1 \]

### Step 2: Find the Adjoint of Matrix A
The adjoint of a 2x2 matrix is given by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements. For our matrix \( A \):
\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \begin{pmatrix} 6 & -5 \\ 7 & -6 \end{pmatrix} \]

### Step 3: Calculate the Inverse of Matrix A
The inverse of a matrix \( A \) can be calculated using the formula:
\[ A^{-1} = \frac{\text{adj}(A)}{\text{det}(A)} \]
Substituting the values we found:
\[ A^{-1} = \frac{1}{-1} \begin{pmatrix} 6 & -5 \\ 7 & -6 \end{pmatrix} = -1 \begin{pmatrix} 6 & -5 \\ 7 & -6 \end{pmatrix} = \begin{pmatrix} -6 & 5 \\ -7 & 6 \end{pmatrix} \]

### Final Answer
Thus, the inverse of the matrix \( A \) is:
\[ A^{-1} = \begin{pmatrix} -6 & 5 \\ -7 & 6 \end{pmatrix} \]

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