Question:
If $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right], B=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$, find adj $(A B)$
Solution:
$A \times B=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$
$A \times B$ is a non $-$ singular matrix. Therefore, it is invertible.
Let $C_{i j}$ be a cofactor of $a_{i j}$ in $A$.
The cofactors of element $A$ are given by
$C_{11}=d$
$C_{12}=-c$
$C_{21}=-b$
$C_{22}=a$
$\therefore \operatorname{adj} A=\left[\begin{array}{cc}d & -c \\ -b & a\end{array}\right]^{T}=\left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]$