The inverse of the matrix $A=\begin{bmatrix} a & b\\c & \frac{1+bc}{a}\end{bmatrix}$ is  :

$\begin{bmatrix} \frac{1+bc}{a} & -b\\-c & a\end{bmatrix}$

The correct answer is Option (2) → $\begin{bmatrix} \frac{1+bc}{a} & -b\\-c & a\end{bmatrix}$

$A=\begin{bmatrix} a & b\\c & \frac{1+bc}{a}\end{bmatrix}$

$|A|=1+bc-bc=1$

finding cofactors

$c_{11}=\frac{1+bc}{a}, c_{12}=-c$

$c_{21}=-b,c_{22}=a$

so $Adj\,A=\begin{bmatrix} c_{11} & c_{12}\\c_{21} & c_{22}\end{bmatrix}^T$

$Adj\,A=\begin{bmatrix} \frac{1+bc}{a} & -b\\-c & a\end{bmatrix}$

$A^{-1}=\frac{Adj\,A}{|A|}=\begin{bmatrix} \frac{1+bc}{a} & -b\\-c & a\end{bmatrix}$