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

Explanation

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

### Step 1: Calculate the Determinant of A
The determinant of a 2x2 matrix \[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \] is given by 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: Find the Adjoint of A
The adjoint of a 2x2 matrix is obtained by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements. For the matrix \( A \):
- Swap \( a \) and \( d \): \( 1 \) and \( 3 \)
- Change the signs of \( b \) and \( c \): \( -1 \) and \( 2 \)

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

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

Calculating the inverse:
\[ A^{-1} = \begin{pmatrix} \frac{3}{5} & \frac{1}{5} \\ -\frac{2}{5} & \frac{1}{5} \end{pmatrix} \]

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

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