**Problem**. Let and be matrices with entries from some field Let be distinct elements in such that are all nilpotent. Prove that and are nilpotent.

**Solution**. Since are nilpotent, their eigenvalues are all zero and hence, by the Cayley-Hamilton theorem, for all Let be an indeterminate. So the equation

has at least roots in We will show that this is not possible unless which will complete the solution. To do so, let’s expand the left hand side in to get

where each is in the -algebra generated by and Let and let and be the -entries of and respectively. Then the -entry of the matrix on the left hand side of is

where are the -entries of So is telling us that the polynomial which has degree at most has at least roots in This is not possible unless Thus