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

Explanation

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

### 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:

\[ a = 1, \quad b = -1, \quad c = 2, \quad d = 3 \]

Calculating the determinant:

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

### Step 2: Check if the Determinant is Non-Zero

Since \( \text{det}(A) = 5 \) which is not equal to zero, the inverse of the matrix exists.

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

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 \( A \):

\[ \text{adj}(A) = \begin{pmatrix} 3 & 1 \\ -2 & 1 \end{pmatrix} \]

### Step 4: Transpose the Adjoint Matrix

The adjoint matrix is already in the required form for a 2x2 matrix, but we can write it explicitly as:

\[ \text{adj}(A) = \begin{pmatrix} 3 & 1 \\ -2 & 1 \end{pmatrix} \]

### Step 5: 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:

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

This results in:

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

### Final Answer

Thus, 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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