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 $\displaystyle A$ and $\displaystyle B$ be a $\displaystyle n\times k$ and $\displaystyle k\times n$ complex matrices. Then,*

$\displaystyle \det\left(I_n+AB\right)=\det\left(I_k+BA\right)$