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?
 A: Applying the formula in this link, we end up with
$$
X := S_A = S_D = D - CA^{-1}B = (aI + S) - (aI + S)^{-1}\\ 
$$
$$
M^{-1} = \pmatrix{
X^{-1} & (aI + S)^{-1}X^{-1}\\
(aI + S)^{-1}X^{-1} & X^{-1}
}
$$
A: With
$$
A =
\left[
\begin{matrix}
aI + S & I \\
I & aI + S
\end{matrix}
\right]
$$
we get
$$
M A =
\left[
\begin{matrix}
aI + S & -I \\
-I & aI + S
\end{matrix}
\right]
\left[
\begin{matrix}
aI + S & I \\
I & aI + S
\end{matrix}
\right]
=
\left[
\begin{matrix}
(aI + S)^2-I & 0 \\
0 & (aI+S)^2 - I
\end{matrix}
\right]
$$
Having
$$
F = ((aI + S)^2 - I)^{-1}
$$
would lead to
$$
B =
\left[
\begin{matrix}
(aI + S)F & F \\
F & (aI + S)F
\end{matrix}
\right]
$$
and
$$
M B =
\left[
\begin{matrix}
aI + S & -I \\
-I & aI + S
\end{matrix}
\right]
\left[
\begin{matrix}
(aI + S)F & F \\
F & (aI + S)F
\end{matrix}
\right]
=
\left[
\begin{matrix}
((aI + S)^2-I)F & 0 \\
0 & F(-I + (aI+S)^2)
\end{matrix}
\right]
=
\left[
\begin{matrix}
I & 0 \\
0 & I
\end{matrix}
\right]
$$
Alas I do not know, if such an $F$ exists in this case. 
$B$ would be an inverse of $M$.
