If $\lambda$ is an eigenvalue of $A$, determine eigenvalues of $A^2$ and $A^3$ Also what is an eigenvalue of $A^n$?
I know you can square the eigenvalues, etc... but how do I prove this for a general case? 

If $E$ is an eigenspace of $A$ correspoinding to an eigenvalue $\lambda$ then
$A$ acts as a multiplication by $\lambda$ operator on $E$, that is,
$A x = \lambda x$ if $x \in E$.
Note that $AE \subset E$, so you can apply $A$ again, where it still acts as a multiplication by $\lambda$ operator on $E$. Hence $A(A x) = A(\lambda x) = \lambda Ax = \lambda ^2 x$.
