Use the Cayley-Hamilton Theorem to find the inverse of this $3$-by-$3$ matrix $A=\begin{pmatrix}
2&0&0\\
1&3&0\\
-3&5&3
\end{pmatrix}.$
I can find the characteristic polynomial $C_a(A)= -A^3 + 8A^2-21A+18$.
How to continue from here to use the Cayley-Hamilton theorem to find the inverse matrix, $A^{-1}$?

${Hint:}$
If $-A^3+8A^2-21A+18I=0$, then $A(-A^2+8A-21I)=-18I$.
