Question:
If adj $A=\left[\begin{array}{cc}2 & 3 \\ 4 & -1\end{array}\right]$ and adj $B=\left[\begin{array}{cc}1 & -2 \\ -3 & 1\end{array}\right]$, find adj $A B$
Solution:
Given:
$a d j A=\left[\begin{array}{cc}2 & 3 \\ 4 & -1\end{array}\right]$
$a d j B=\left[\begin{array}{cc}1 & -2 \\ -3 & 1\end{array}\right]$
For any two non-singular matrices,
$a d j(A B)=(a d j B) \times(a d j A)$
$\Rightarrow \operatorname{adj}(A B)=\left[\begin{array}{cc}-6 & 5 \\ -2 & -10\end{array}\right]$