Consider the $k \times k$ block matrix:

$$C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ B & B & B & \cdots & A \end{array}\right) = I_k \otimes (A - B) + \mathbb{1}_k \otimes B$$

where $A$ and $B$ are size $n \times n$ and $\mathbb{1}$ is the matrix of all ones.

It would seem that the formula for the determinant of $C$ is simply:

$$\det(C) = \det(A-B)^{k-1} \det(A+(k-1) B)$$

Can anyone explain why this seems to be true or offer a proof or direct me to a proof?