relatively prime

Let be a linear transform on a vector field . Suppose is the minimal polynomial of , is a polynomial. Show that if is invertible, then is relatively prime. That is,

You know that the minimal polynomial is irreducible. So, if was not one, this would say that which would imply that , yeah?

Why minimal polynomial is irreducible? It of course could be reducible? Do you see the Jordan canonical form of a matrix...

My mistake, I was being silly. The key is that the eigenvalues of are where are the eigenvalues of . Now, note that since is invertible it has no zero eigenvalues, and so for all --I think you can conclude from that.