<s>[INST] Determinant of specially structured block matrix How do you compute the determinant of the following $nd \times nd$ block matrix?
$$M = \begin{bmatrix}A+B & A & A & \dots & A & A\\ A & A+B & A & \dots & A & A\\ A & A & A+B & \dots & A & A\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ A & A & A & \dots & A+B & A\\ A & A & A & \dots & A & A+B\end{bmatrix}$$
where $A$ and $B$ are $d \times d$ matrices.
 [/INST] It has the same determinant as the matrix
$$
\begin{pmatrix}
B & 0 & \cdots & -B \\
0 & B & \cdots & -B \\
\vdots & \vdots & \ddots & \vdots\\
A & A & \cdots & A+B
\end{pmatrix},
$$
which has the same determinant as
$$
\begin{pmatrix}
B & 0 & \cdots & 0 \\
0 & B & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots\\
A & A & \cdots & nA+B
\end{pmatrix},
$$
which is easy to compute.
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