Write `A^(-1)` for `A=[{:(2,5),(1,3):}]`.

Answer

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

### Step 1: Calculate the Determinant of 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 \( A \):
- \( a = 2 \)
- \( b = 5 \)
- \( c = 1 \)
- \( d = 3 \)

Calculating the determinant:
\[ \text{det}(A) = (2)(3) - (5)(1) = 6 - 5 = 1 \]

### Step 2: Check if the Determinant is Non-Zero
Since \( \text{det}(A) = 1 \), which is non-zero, the inverse of \( A \) exists.

### Step 3: Find the Adjoint of A
The adjoint of a 2x2 matrix is found by swapping the elements on the main diagonal and changing the signs of the elements on the other diagonal. The adjoint of \( A \) is given by:
\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \begin{pmatrix} 3 & -5 \\ -1 & 2 \end{pmatrix} \]

### Step 4: 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:
\[ A^{-1} = \frac{1}{1} \cdot \begin{pmatrix} 3 & -5 \\ -1 & 2 \end{pmatrix} = \begin{pmatrix} 3 & -5 \\ -1 & 2 \end{pmatrix} \]

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

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