Find the inverse of each of the following matrice :
`[{:(2,-2),(4,3):}]`

Answer

To find the inverse of the matrix \( A = \begin{pmatrix} 2 & -2 \\ 4 & 3 \end{pmatrix} \), 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 \( A \):
- \( a = 2 \)
- \( b = -2 \)
- \( c = 4 \)
- \( d = 3 \)

So, the determinant is:
\[ \text{det}(A) = (2)(3) - (-2)(4) = 6 + 8 = 14 \]

### Step 2: Calculate the Adjoint of Matrix A
The adjoint of a 2x2 matrix is found by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements. For our matrix \( A \):
\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \begin{pmatrix} 3 & 2 \\ -4 & 2 \end{pmatrix} \]

### 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:
\[ A^{-1} = \frac{1}{14} \begin{pmatrix} 3 & 2 \\ -4 & 2 \end{pmatrix} \]
This results in:
\[ A^{-1} = \begin{pmatrix} \frac{3}{14} & \frac{2}{14} \\ -\frac{4}{14} & \frac{2}{14} \end{pmatrix} = \begin{pmatrix} \frac{3}{14} & \frac{1}{7} \\ -\frac{2}{7} & \frac{1}{7} \end{pmatrix} \]

### Final Answer
The inverse of the matrix \( A \) is:
\[ A^{-1} = \begin{pmatrix} \frac{3}{14} & \frac{1}{7} \\ -\frac{2}{7} & \frac{1}{7} \end{pmatrix} \]