Inversion of a block matrix Let $S$ to be a symmetric and positive semi-definite matrix of size $n$. What is the inverse of the following block matrix
$$
M_{2n\times 2n}=
\begin{bmatrix}
aI+S & -I\\ -I & aI+S
\end{bmatrix}
$$
where $a$ is an arbitrary real scalar?

Applying the formula in this link, we end up with
$$
X := S_A = S_D = D - CA^{-1}B = (aI + S) - (aI + S)^{-1}\\ 
$$
$$
M^{-1} = \pmatrix{
X^{-1} & (aI + S)^{-1}X^{-1}\\
(aI + S)^{-1}X^{-1} & X^{-1}
}
$$
