If $C = A + B$ and $D = A - B$,
and $A$ and $B$ are $n\times n$ and $D , C$ are invertible,
prove the following
$$\begin{bmatrix}A & B \\B & A\end{bmatrix}^{-1} = \frac12 \begin{bmatrix}C^{-1} + D^{-1} & C^{-1} - D^{-1} \\ C^{-1} + D^{-1} & C^{-1} - D^{-1}\end{bmatrix}$$
I tried multiplying them I failed (maybe you can),
I did some other work and did not made any thing better.
Glad if you can help.