Find the inverse of each of the following matrice :
`[{:(-1,5),(-3,2):}]`

Answer

To find the inverse of the matrix \( A = \begin{pmatrix} -1 & 5 \\ -3 & 2 \end{pmatrix} \), we will follow these steps:

### 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 \( A \):
- \( a = -1 \)
- \( b = 5 \)
- \( c = -3 \)
- \( d = 2 \)

Calculating the determinant:
\[ \text{det}(A) = (-1)(2) - (5)(-3) = -2 + 15 = 13 \]

### Step 2: Calculate the Adjoint of Matrix A
The adjoint of a 2x2 matrix is given by:
\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]
For our matrix \( A \):
- \( d = 2 \)
- \( -b = -5 \)
- \( -c = 3 \)
- \( a = -1 \)

Thus, the adjoint of \( A \) is:
\[ \text{adj}(A) = \begin{pmatrix} 2 & -5 \\ 3 & -1 \end{pmatrix} \]

### Step 3: Calculate the Inverse of Matrix A
The inverse of matrix \( A \) is given by the formula:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
Substituting the values we calculated:
\[ A^{-1} = \frac{1}{13} \cdot \begin{pmatrix} 2 & -5 \\ 3 & -1 \end{pmatrix} \]

### Step 4: Write the Final Result
Thus, the inverse of matrix \( A \) is:
\[ A^{-1} = \begin{pmatrix} \frac{2}{13} & \frac{-5}{13} \\ \frac{3}{13} & \frac{-1}{13} \end{pmatrix} \]