at least one of M or N is invertible. i'll assume that is invertible. (a similar proof works for M) first a Lemma which is always true:
Proof: clearly thus for any positive integer we have:
so if then:
now if N is invertible, then by the Lemma: so: for any positive integer
it's obvious now that if is nilpotent, then is nilpotent too. Q.E.D.