find the inverse of the following matrix : `[{:(1,-1),(2,3):}]`

Explanation

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

### Step 1: Calculate the Determinant of the Matrix
The determinant of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated using the formula:
\[ \text{det}(A) = ad - bc \]
For our matrix:
- \( a = 1 \)
- \( b = -1 \)
- \( c = 2 \)
- \( d = 3 \)

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

### Step 2: Check if the Determinant is Non-Zero
Since the determinant \( \text{det}(A) = 5 \) is not equal to zero, the inverse of the matrix exists.

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

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

### Step 4: Calculate the Inverse of the Matrix
The inverse of the matrix \( A \) is given by the formula:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
Substituting the values we found:
\[ A^{-1} = \frac{1}{5} \cdot \begin{pmatrix} 3 & 1 \\ -2 & 1 \end{pmatrix} \]
This results in:
\[ A^{-1} = \begin{pmatrix} \frac{3}{5} & \frac{1}{5} \\ -\frac{2}{5} & \frac{1}{5} \end{pmatrix} \]

### Final Answer
The inverse of the matrix \( A \) is:
\[ A^{-1} = \begin{pmatrix} \frac{3}{5} & \frac{1}{5} \\ -\frac{2}{5} & \frac{1}{5} \end{pmatrix} \]