I have a block matrix and its inverse:
$$M=\pmatrix{A&B\\C&D}, M^{-1}=\pmatrix{E&F\\G&H}$$
I want to compute $A^{-1}$ from $M^{-1}$ ideally without involving any other inversion operations. As far as I can tell this seems infeasible. What I have tried is writing out the equations:
$$MM^{-1}=\pmatrix{AE+BG&AF+BH\\CE+DG&CF+DH} = \pmatrix{I&0\\0&I}$$ $$M^{-1}M=\pmatrix{EA+FC&EB+FD\\GA+HC&GB+HD} = \pmatrix{I&0\\0&I}$$
But all of those result in some inverse when I try to compute $A^{-1}$. Can someone confirm that this is indeed infeasible? If this helps, I can constrain this to $M$ and $M^{-1}$ being real symmetric matrices.
Just because $M$ is invertible, $A$ doesn’t have to. Consider for example the $2\times 2$-matrix $$M=\begin{bmatrix}0&1\\1&0\end{bmatrix}$$
I didn’t yet think about what happens, if we assume $A$ to be invertible though.