linear algebra - Determinant of block matrices

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?