this problem has been bugging me for hours! i don't know the answer for the general case, but i can prove that the answer is "yes" if

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:

Lemma:

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.