I've been brushing up on some linear algebra this summer, and came across this problem in a practice book. I'm totally stuck on how to do this!

Prove that if M is an nxn matrix and can be written as

where A and C are square matrices, then det(M) = det(A)*det(C).

Anyone have a proof for this? I was almost wondering if I should try and use induction for n>=2 since this is true for a 2x2 matrix. Is there an easier (or more correct) way to prove this?