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 $0,2,-4$.
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!