Specifically I am interested in the entries of the inverse of
$$ \mathcal{A}= \begin{bmatrix} A & B & & \\ B & \ddots & \ddots & \\ & \ddots & & B \\ & & B & A \\ \end{bmatrix} $$
where $B$ and $A$ are square invertable matrices. I am going though the algebra by trying to find $Z$ such that $\mathcal{A}Z = \mathbb{I}$.
This algebra is very tedious and so I was wondering if such an expression already exists for such a matrix.