Inversion of a block matrix

Question

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?

Answer 1

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

Answer 2

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 $M$.