Question:
Write $A^{-1}$ for $A=\left[\begin{array}{ll}2 & 5 \\ 1 & 3\end{array}\right]$
Solution:
$|A|=\left|\begin{array}{ll}2 & 5 \\ 1 & 3\end{array}\right|=1 \neq 0$
Let $C_{i j}$ be the cofactor of $a_{i j}$ in $A$.
The cofactors of element $A$ are given by
$C_{11}=3$
$C_{12}=-1$
$C_{21}=-5$
$C_{22}=2$
$\operatorname{adj} A=\left[\begin{array}{cc}3 & -1 \\ -5 & 2\end{array}\right]^{T}=\left[\begin{array}{cc}3 & -5 \\ -1 & 2\end{array}\right]$
$|A|=6-5=1$
$\therefore A^{-1}=\frac{1}{|A|}$ adj $A=\left[\begin{array}{cc}3 & -5 \\ -1 & 2\end{array}\right]$