How to find the eigenvectors of $(I + A)^{100}$ 



How to find the eigenvectors of $(I + A)^{100}$
, given $A=\begin{bmatrix}-4 & -5\\10 & 11 \end{bmatrix}$.
I've tried to solve this by the determinant, like this
$det(I+A)=1+|Ai\rm|$, where $Ai\rm$ are all principal submatrices of $A$.
 but I can't get more idea about this. If anyone could help would be much appreciated. Thanks.



Hint:


*

*If $Ax = \lambda x$, then what is $(A+I)x$ equal to?

*If $Bx=\lambda x$, then what is $B^2x$ equal to? (remember, $B^2x=B(Bx)$!)

*From (2), if $Bx=\lambda x$, then what is $B^{100}x$ equal to?


Now, join (1) and (3) together and you should find a simple connection between eigenvectors of $A$ and eigenvectors of $(A+I)^{100}$.
