Matrices & Systems · Challenge problem

Let $B = \begin{pmatrix}1 & 2\\3 & 4\end{pmatrix}$ and let $M$ be the $4\times 4$ block matrix $M = \begin{pmatrix}I & B\\0 & I\end{pmatrix}$, where $I$ is the $2\times 2$ identity and $0$ the $2\times 2$ zero matrix. Find the sum of all sixteen entries of $M^{-1}$.