Well, I can think of two possible approaches to this problem. One is to diagonalize F. I think you might be able to do that, since it is symmetric. That would enable you to produce a closed-form expression for the required power of F.

Another approach is to reason about the problem and come up with an argument (I don't know what that would be, but it seems to me that you might be able to come up with one) showing the desired result.