Consider the matrix `A=[{:(2,3),(4,5):}]`.
Hence , find `A^(-1)`.

Answer

To find the inverse of the matrix \( A = \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix} \), we can 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 calculated using the formula:
\[ \text{det}(A) = ad - bc \]
For our matrix \( A \):
- \( a = 2 \)
- \( b = 3 \)
- \( c = 4 \)
- \( d = 5 \)

Thus, the determinant is:
\[ \text{det}(A) = (2)(5) - (3)(4) = 10 - 12 = -2 \]

### Step 2: Find the Adjoint of 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 \( A \):
- \( d = 5 \)
- \( -b = -3 \)
- \( -c = -4 \)
- \( a = 2 \)

Thus, the adjoint is:
\[ \text{adj}(A) = \begin{pmatrix} 5 & -3 \\ -4 & 2 \end{pmatrix} \]

### Step 3: Calculate the Inverse of 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 found:
\[ A^{-1} = \frac{1}{-2} \cdot \begin{pmatrix} 5 & -3 \\ -4 & 2 \end{pmatrix} \]
This simplifies to:
\[ A^{-1} = \begin{pmatrix} \frac{5}{-2} & \frac{-3}{-2} \\ \frac{-4}{-2} & \frac{2}{-2} \end{pmatrix} = \begin{pmatrix} -\frac{5}{2} & \frac{3}{2} \\ 2 & -1 \end{pmatrix} \]

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

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