Find the inverse of each of the matrices (if it exists).

Let $A=\left[\begin{array}{lll}2 & 1 & 3 \\ 4 & -1 & 0 \\ -7 & 2 & 1\end{array}\right]$.

We have,

$\begin{aligned}|A| &=2(-1-0)-1(4-0)+3(8-7) \\ &=2(-1)-1(4)+3(1) \\ &=-2-4+3 \\ &=-3 \end{aligned}$

Now,

$A_{11}=-1-0=-1, A_{12}=-(4-0)=-4, A_{13}=8-7=1$

$A_{21}=-(1-6)=5, A_{22}=2+21=23, A_{23}=-(4+7)=-11$

$A_{31}=0+3=3, A_{32}=-(0-12)=12, A_{33}=-2-4=-6$

$\therefore \operatorname{adj} A=\left[\begin{array}{ccc}-1 & 5 & 3 \\ -4 & 23 & 12 \\ 1 & -11 & -6\end{array}\right]$

$\therefore A^{-1}=\frac{1}{|A|}$ adjA $=-\frac{1}{3}\left[\begin{array}{lll}-1 & 5 & 3 \\ -4 & 23 & 12 \\ 1 & -11 & -6\end{array}\right]$