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

Let $A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1\end{array}\right]$

We have,

$|A|=1(-3-0)-0+0=-3$

Now,

$A_{11}=-3-0=-3, A_{12}=-(-3-0)=3, A_{13}=6-15=-9$

$A_{21}=-(0-0)=0, A_{22}=-1-0=-1, A_{23}=-(2-0)=-2$

$A_{31}=0-0=0, A_{32}=-(0-0)=0, A_{33}=3-0=3$

$\therefore \operatorname{adj} A=\left[\begin{array}{ccc}-3 & 0 & 0 \\ 3 & -1 & 0 \\ -9 & -2 & 3\end{array}\right]$

$\therefore A^{-1}=\frac{1}{|A|}$ adj $A=-\frac{1}{3}\left[\begin{array}{ccc}-3 & 0 & 0 \\ 3 & -1 & 0 \\ -9 & -2 & 3\end{array}\right]$