Thread: Determinental Identity

The challenge section has been pretty dead as of late. I think one of the reasons is that some of the problems (or at least the problem's subject matter) is somewhat inaccessible. So, I propose the following question which has the threefold benefit of a) having an easy to understand statement, b) having both a difficult but elementary and an easier but less elementary solution, and c) having real applications in matrix based mathematics. So:


Problem: Let and be a and complex matrices. Then,

We have a more general result: if we denote by (resp. ) the characteristic polynomial of (resp ), we have . To see that, we notice that

and take the determinant.
Now, we take to show the asked result.