$A$ is a tridiagonal matrix and $B$ is an anti-tridiagonal matrix, both of size $n\times n$, such that $B^2$ is a diagonal matrix. Is it possible to express $\det(A+B)$ in terms of $\det(A),\det(B),n$ and the eigenvalues of $A$ and $B$?
In fact, any clue to get the spectrum of $A+B$ will also help. I know that the Courant-Weyl inequalities give bounds of the eigenvalues of $A+B$, but obtaining only the bounds are not sufficient for my problem.
Could someone please help me? Thanks in advance.