TITLE: Compute the determinant of this block matrix
QUESTION [1 upvotes]: Let the following block matrix $$F= \begin{bmatrix}
    A_{11} & \cdots & A_{1L} \\ \vdots & & \vdots \\
    A_{L1} & \cdots & A_{LL} 
\end{bmatrix} \in \mathbb{Z}^{nL \times nL}$$
where each matrix $A_{ij} \in \mathbb{Z}^{n \times n}$ is defined as
$$A_{ij} = R \cdot X \cdot Y_{ij} \cdot R^{T}$$
$\forall i,j = \overline{1,L}$, for some matrix $R \in \mathbb{Z}^{n \times n}$ and some $n \times n$ diagonal matrices $X$, $Y_{ij}$.
Prove: $$\det(F) = \det(R)^{2L} \cdot \det(X)^{L} \cdot \det\left(\begin{bmatrix}
    Y_{11} & \cdots & Y_{1L} \\ \vdots & & \vdots \\
    Y_{L1} & \cdots & Y_{LL} 
\end{bmatrix}\right)$$ 
Any hints how to start to compute this determinant would be appreciated.

REPLY [1 votes]: Hint: (empty blocks are zeros)
$$
F=\begin{bmatrix}R&&&\\&R&&\\&&\ddots&\\&&&R\end{bmatrix}
\begin{bmatrix}X&&&\\&X&&\\&&\ddots&\\&&&X\end{bmatrix}Y
\begin{bmatrix}R^T&&&\\&R^T&&\\&&\ddots&\\&&&R^T\end{bmatrix}.
$$
{"set_name": "stack_exchange", "score": 1, "question_id": 2271396}