The following questions is apprently an extentsion of cayley hamilton and I was not sure if that meant we where supposed to use the proof of said result but I do not know how to extend the proof using Hofman and Kunze's text.
Suppose that $p_M$ is the characteristic polynomial for a matrix $M$ with entries in a field. Suppose there exists a matrix $N$ such that $MN = NM$.
How do we show that $ p_M (N) = A (M - N) $ for some matrix $A$ such that $A$ commutes with $M$ and $N$?