Matrices & Systems · Challenge problem

Let $N = \begin{pmatrix}0 & 1 & 0\\0 & 0 & 1\\0 & 0 & 0\end{pmatrix}$. A $3\times 3$ matrix $A$ commutes with $N$ (that is, $AN = NA$), and the third column of $A$ is $\begin{pmatrix}5\\3\\2\end{pmatrix}$. Find the sum of all nine entries of $A^{-1}$.