We can easily show it holds for the case of matrixes P of , however, it doesn't hold in general ( take you'll find that one of the matrixes has an additional eigenvalue)
Let's see the 2x2 matrixes. Remember that the characteristic polynomial of a matrix in this case is given by:
For any 2 matrixes and we have:
So if is a 2x2 matrix we have: and and therefore for all .
Thus the assertion is valid when P is a 2x2 matrix