If `{:A=[(3,1),(-1,2)]:}," then "A^(-2)=`

Options

A. `{:[(8,-5),(-5,3)]:}`
B. `{:[(8,-5),(5,3)]:}`
C. `{:[(8,-5),(-5,-3)]:}`
D. `{:[(8,5),(-5,3)]:}`

Answer

To find \( A^{-2} \) for the matrix \( A = \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate \( A^2 \)

To find \( A^2 \), we multiply matrix \( A \) by itself:

\[ A^2 = A \cdot A = \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix} \cdot \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix} \]

Calculating the elements of \( A^2 \):

- First row, first column: \( 3 \cdot 3 + 1 \cdot (-1) = 9 - 1 = 8 \)
- First row, second column: \( 3 \cdot 1 + 1 \cdot 2 = 3 + 2 = 5 \)
- Second row, first column: \( -1 \cdot 3 + 2 \cdot (-1) = -3 - 2 = -5 \)
- Second row, second column: \( -1 \cdot 1 + 2 \cdot 2 = -1 + 4 = 3 \)

Thus, we have:

\[ A^2 = \begin{pmatrix} 8 & 5 \\ -5 & 3 \end{pmatrix} \]

### Step 2: Calculate the Determinant of \( A^2 \)

To find \( A^{-2} \), we need the determinant of \( A^2 \):

\[ \text{det}(A^2) = (8)(3) - (5)(-5) = 24 + 25 = 49 \]

### Step 3: Calculate the Adjoint of \( A^2 \)

The adjoint of a \( 2 \times 2 \) matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by \( \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \).

For \( A^2 = \begin{pmatrix} 8 & 5 \\ -5 & 3 \end{pmatrix} \):

\[ \text{adj}(A^2) = \begin{pmatrix} 3 & -5 \\ 5 & 8 \end{pmatrix} \]

### Step 4: Calculate \( A^{-2} \)

Now we can find \( A^{-2} \) using the formula:

\[ A^{-2} = \frac{1}{\text{det}(A^2)} \cdot \text{adj}(A^2) \]

Substituting the values we found:

\[ A^{-2} = \frac{1}{49} \cdot \begin{pmatrix} 3 & -5 \\ 5 & 8 \end{pmatrix} \]

Thus, we have:

\[ A^{-2} = \begin{pmatrix} \frac{3}{49} & \frac{-5}{49} \\ \frac{5}{49} & \frac{8}{49} \end{pmatrix} \]

### Final Answer

\[ A^{-2} = \begin{pmatrix} \frac{3}{49} & \frac{-5}{49} \\ \frac{5}{49} & \frac{8}{49} \end{pmatrix} \]