For the determinant of $\bar{\mathcal{A}} $ there is formula in this paper actually arxiv.org/pdf/0712.0681.pdf

I was looking for some expressions that I can use in a proof... Would you happen to know if expressions exist for the block entries of the inverse of a matrix of the form $\bar{\mathcal{A}}$ and the determinant of $\bar{\mathcal{A}}$?

Thank you for your help thus far. Would it be possible to expand on the second and the third point a little? I am working through your solution and have made edit to show my progress. It is the case that if $A = QSZ^{*}$ then $S$ is diagonal if $A$ is tridiagonal? Also I would have to find the matrices $Q,S,Z$ explicitly if I was to continue with this method? ( the same goes for B)

The blocks are invertiable, they are tridiagonal and teoeplitz. This is a lot of extra information, I should have included it! I will update the post accordingly.