In the context of Quantum Many Body problem, I have encountered the following determinant:
$$\mathbb{F}[\{\mathbb{A}_{i}^{}\},\mathbb{B}_{}^{}]=\det[\mathbb{I}_{}^{} - \mathbf{A}_{}^{}\cdot\mathbf{B}_{}^{}],$$
where,
$\mathbb{I}_{}^{}$ is the $nN \times nN$ identity matrix,
$\mathbf{A}_{}^{}$ is an $nN \times nN$ complex block diagonal matrix defined as, $$\mathbf{A}_{}^{} = \text{Diag}\begin{pmatrix}\mathbb{A}_{1}^{} & \cdots & \mathbb{A}_{N}^{}\end{pmatrix},$$ where $\mathbb{A}_{i}^{}$ are $n \times n$ general complex matrices and
$\mathbf{B}_{}^{}$ is an $nN \times nN$ complex matrix defined as, $$\mathbf{B}_{}^{} = \mathbf{1}\otimes \mathbb{B}_{}^{},$$ where $\mathbb{B}_{}^{}$ is a $n \times n$ general complex matrix and $\mathbf{1}$ is the $N \times N$ 'one' matrix defined as, $$\mathbf{1} = \begin{pmatrix} 1 & \cdots & 1\end{pmatrix}_{}^{T}\begin{pmatrix} 1 & \cdots & 1\end{pmatrix}.$$
Is it possible to write an explicit form for $\mathbb{F}[\{\mathbb{A}_{i}^{}\},\mathbb{B}_{}^{}]$?
$\textbf{Some thoughts:}$
(i) Is there any block matrix variant of matrix-determinant lemma? Can writing $\mathbf{A}\cdot\mathbf{B}==\begin{pmatrix}\mathbb{A}_{1}^{T} & \cdots & \mathbb{A}_{N}^{T}\end{pmatrix}_{}^{T}\begin{pmatrix} \mathbb{B}_{}^{} & \cdots & \mathbb{B}_{}^{}\end{pmatrix}$ help?
(ii) Is it possible to define 'eigenvalue-matrix' problem and express the needed in terms of determinants of $n \times n$ eigenmatrices?
As you have noted, we indeed have $$ \mathbf{A \cdot B} = \overbrace{\pmatrix{A_1 \\ \vdots \\ A_N}}^P \overbrace{\pmatrix{B & \cdots & B}}^Q. $$ With that, we can use the WA-identity (which is indeed a generalization of the matrix determinant lemma) to get $$ \det(I_{nN} - P \cdot Q) = \det(I_n - Q \cdot P) = \det[I_n - B(A_1 + \cdots + A_N)]. $$