I have a $3n\times3n$ matrix $M$ that is in the following block tridiagonal form: $$M=\begin{pmatrix} A & B^T & 0\\ B & A & UBU \\ 0 & UB^T U & A\\ \end{pmatrix}$$ where $A,B,U$ are $n\times n$ real matrices. I also know that $A$ and $U$ are symmetric and $U$ is orthogonal (it is the exchange matrix), and that $\det{B} = 0$. I want to know if a closed-form analytical expression for the determinant of $M$ exists.
What I have tried so far:
From the paper mentioned in this post, I found that one can write
$$\det{M} = \det{\Lambda_1}\det{\Lambda_2}\det{\Lambda_3},$$ with $\Lambda_1 = A$, $\Lambda_2 = A - B \Lambda_1^{-1} B^T$, and $\Lambda_3 = A - (UB^TU)\Lambda_2^{-1}(UBU)$.
I have not been able to progress beyond this point as I do not know of some clever way to do the inversions.
Thanks!
If A is invertible then by the determinant of a block matrix, $\det(M)=\det(A)\det(A)\det(A-B^TA^{-1}B-UB^TUA^{-1}UBU)$