I am having the following block matrix
$$\left[\begin{array}{cccc}
\mathbf{A+B} & \mathbf{B} & \cdots & \mathbf{B} \\
\mathbf{B} & \mathbf{A+B} & \cdots & \mathbf{B} \\
\vdots & \vdots & \ddots & \vdots\\
\mathbf{B} & \mathbf{B} & \cdots & \mathbf{A+B}
\end{array}\right]$$
where $\mathbf{A}$ and $\mathbf{B}$ are full rank, symmetric square matrices. There are $n$ blocks in each direction. I want to obtain the determinant of the block matrix.
I play with some examples and the determinants seems to be
$$\det(\mathbf{A})^{n-1}\det(\mathbf{A}+n\mathbf{B})$$
May I ask whether this is correct or not, and is there any proof?
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