TITLE: Determinant of block tridiagonal matrices
QUESTION [5 upvotes]: Is there a formula to compute the determinant of block tridiagonal matrices when the determinants of the involved matrices are known?
In particular, I am interested in the case
$$A = \begin{pmatrix} J_n & I_n & 0 & \cdots & \cdots & 0 \\ I_n & J_n & I_n & 0 & \cdots & 0 \\ 0 & I_n & J_n  & I_n & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots &  \ddots & 0 \\ 0 & \cdots & \cdots & I_n & J_n & I_n \\ 0 & \cdots & \cdots & \cdots & I_n & J_n \end{pmatrix}$$
where $J_n$ is the $n \times n$ tridiagonal matrix whose entries on the sub-, super- and main diagonals are all equal to $1$ and $I_n$ is identity matrix of size $n$.
I have asked this question before on MathStackExchange, where a user came up with an algorithm. Nevertheless, I am interested if there is an explicit formula (or at least, if one can say in which cases the determinant is nonzero).

REPLY [5 votes]: The Kronecker product idea brought up in Algebraic Pavel's comment on the original maths stack exchange question seems like a good way to approach the particular case of interest to you. Specifically, assuming $A$ is $m n \times m n$, i.e., there are $m$ block rows and columns, then
$$A = J_m \otimes I_n + I_m \otimes J_n - I_{mn},$$
and the $mn$ eigenvalues of $A$ are given by
$$\lambda_{ij} = \Big(1+2 \cos \frac{i \pi}{m+1}\Big) + \Big(1+2 \cos \frac{j \pi}{n+1}\Big) - 1, \qquad 1 \le i \le m, 1 \le j \le n.$$
(I used the formula for the eigenvalues of the $J$ matrices from Denis Serre's answer here.) The determinant is then
$$\det A = \prod_{i=1}^m \prod_{j=1}^n \lambda_{ij}.$$
If you're only after characterizing when $A$ is singular, then you need only determine when any of the $\lambda_{ij}$ can be zero, which looks fairly straightforward.