I'm working through the Doubly Stochastic Matrices section of Linear Algebra in Action for a project and am a bit stuck on Exercise 23.18. The problem goes as follows.
Let $A=\begin{bmatrix} B&0\\C&D\end{bmatrix}$ where $B$ and $D$ are square matrices. Show that $$per(A)=per(B)\cdot per(D).$$ I've convinced myself it should be true by working through a few examples but I'm not sure how to prove it. It shouldn't be too hard but I cant for the life of me figure it out. Maybe it's because I am up far too late. Any help would be much appreciated. Thanks!