Characteristic polynomials of powers and sums of matrices If I know the characteristic polynomial of a matrix $A$, what can I know about the charpoly of $A^2$? And if I have the charpolys of $A$ and $B$, what can I know about the charpoly of $A+B$? I'm trying to solve the following problem:

The eigenvalues of $A$ are $1,-3,0$. Show that the eigenvalues of $A^2+A-2I$ are $0,2,-4$.

Thank you!
Edit: I now know that the eigenvalues of $A^2$ are the squares of the eigenvalues of $A$. I still need help solving the problem. Thanks!
 A: If the eigenvalues of $A$ are $c_1, \ldots, c_n$ then the eigenvalues of $A^k$ are
$c_1^k, \ldots, c_n^k$. You can see this putting $A$ in Jordan form and using the fact that the diagonal entries of a power of a triangular matrix T are the powers of the diagonal entries of that matrix T. 
More generally, if $p$ is a polynomial then the eigenvalues of $p(A)$ are
$p(c_1), \ldots, p(c_n)$ by the same reasoning above.
A: Given a matrix $A$ with eigenvalue $\lambda$, and any polynomial $p(X) = \sum_{k=0}^n a_k X^k$, you can show that $p(\lambda)$ is an eigenvalue of $p(A) = \sum_{k=0}^n a_k A^k$.
Just take an eigenvector $v \ne 0$ of $A$ and look at what $p(A) v$ is.
A: For the eigenvalues: If $Ax= \lambda x$, then $ A^2x = A\lambda x = \lambda^2x $. Does that help you?
Edit: I see that you still have problems, so I will find one of the eigenvalues. Let $x$ be the eigenvector correpsonding to the eigenvalue 1. Then
$$
(A^2 + A - 2I)x = A^2x + Ax - 2Ix = 1^2x + 1x - 2x = (1+1-2)x = 0x,
$$
So $0$ is an eigenvalue of $A^2 + A - 2I$. The other eigenvalues are -2 and 4. Can you prove that?
