Let be the ring of matrices with entries from the field of complex numbers.

Let and suppose that is nilpotent, i.e. for some integer Then But what can we say about Is it equal to Not necessarily, of course! For example, consider

Then but See that in this example,

**Problem**. Let and suppose that is nilpotent. Show that if commute, i.e. then

**Solution**. since is algebraically closed and commute, are simultaneously triangularizable, i.e. there exists an invertible element such that both and are triangular. Since is both nilpotent and triangular, all its diagonal entries are zero and so the diagonal entries of are the same as the diagonal entries of Thus

because are both triangular and the determinant of a triangular matrix is the product of its diagonal entries. So