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

Therefore,

$A^{-1}$ exists

Find adj A:

$A_{11}=+\left|\begin{array}{cc}3 & 0 \\ 2 & -1\end{array}\right|=-3 \quad A_{21}=-\left|\begin{array}{rr}0 & 0 \\ 2 & -1\end{array}\right|=0 \quad A_{31}=+\left|\begin{array}{ll}0 & 0 \\ 3 & 0\end{array}\right|=0$

$A_{12}=-\left|\begin{array}{cc}3 & 0 \\ 5 & -1\end{array}\right|=3 \quad A_{22}=+\left|\begin{array}{cc}1 & 0 \\ 5 & -1\end{array}\right|=-1 \quad A_{32}=-\left|\begin{array}{ll}1 & 0 \\ 3 & 0\end{array}\right|=0$

$A_{13}=+\left|\begin{array}{ll}3 & 3 \\ 5 & 2\end{array}\right|=-9 \quad A_{23}=-\left|\begin{array}{ll}1 & 0 \\ 5 & 2\end{array}\right|=-2 \quad A_{33}=+\left|\begin{array}{ll}1 & 0 \\ 3 & 3\end{array}\right|=3$

adj. $A=\left[\begin{array}{lll}A_{11} & A_{21} & A_{31} \\ A_{12} & A_{22} & A_{32} \\ A_{13} & A_{23} & A_{33}\end{array}\right]=\left[\begin{array}{ccc}-3 & 0 & 0 \\ 3 & -1 & 0 \\ -9 & -2 & 3\end{array}\right]$

As we know, formula to find matrix inverse is:

$\mathrm{A}^{-1}=\frac{1}{|\mathrm{~A}|} \mathrm{adj} \cdot \mathrm{A}$

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