Using the fact that the characteristic polynomial of an nxn matrix is a monic polynomial with coefficients that can be expressed in terms of it's trace and the traces of it's powers upto the power n we get that

as all the traces are 0.

Then the result follows from the Cayley Hamilton Theorem.

I couldn't think of a way of proving it that didnt rely on these two results but I'm sure there is a nice way.

thanks, pomp.