Find the inverse of the matrix`A=[[a, b],[ c,(1+b c)/a]]` and show that `a A^(-1)=(a^2+b c+1)I-a A` .

Explanation

To find the inverse of the matrix \( A = \begin{pmatrix} a & b \\ c & \frac{1 + bc}{a} \end{pmatrix} \) and to show that \( a A^{-1} = (a^2 + bc + 1)I - aA \), we will follow these steps:

### Step 1: Calculate the determinant of matrix \( A \)

The determinant of a 2x2 matrix \( \begin{pmatrix} p & q \\ r & s \end{pmatrix} \) is given by \( ps - qr \).

For our matrix \( A \):
\[
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