I've been thinking about the way that eigenvalues appear on the diagonal of a diagonalized matrix, and found a nice question on it in my textbook:
Prove that if a matrix $A \in C_n$ with $n$ distinct eigenvalues commutes with two matrices $M$ and $N$, then $M$ commutes with $N$.
I suspect the proof must be done using the fact that a matrix with $n$ distinct eigenvalues is diagonalizable, but I have always had issues when it comes to those sorts of algebra manipulations. Any ideas? I'm lost one any other approaches.