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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.

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  • $\begingroup$ Try thinking of $R$ and $X$ as scalars, and use the linearity of the determinant. $\endgroup$ – Itay4 May 8 '17 at 12:36
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Hint: (empty blocks are zeros) $$ F=\begin{bmatrix}R&&&\\&R&&\\&&\ddots&\\&&&R\end{bmatrix} \begin{bmatrix}X&&&\\&X&&\\&&\ddots&\\&&&X\end{bmatrix}Y \begin{bmatrix}R^T&&&\\&R^T&&\\&&\ddots&\\&&&R^T\end{bmatrix}. $$

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