For which power of a matrix does it has eigenvectors?

Question

I Have the following question in my exercise:

Let $T:\Bbb R^2 \rightarrow \Bbb R^2$

$[T]_E= \left[ \begin{matrix} 2&-4\\5&-2\\\end{matrix} \right]$

The question asks to find all $n \in \Bbb N$ so that $T^n$ has a non-trivial invariant subspace.

I Understand that since $\dim(V)=2$ the invariant subspaces which are not trivial (not $\{0\}$ or $\Bbb R^2$) must be of $\dim=1$ and thus be the Eigenvectors of $[T^n]_E$ but I don't get how am I suppose to know for which $n$ the eigenvectors exist.

Answer 1

Hint : If you have $Ax=\lambda x$ for some eigenvector $x$, you also have $A^n x=\lambda^n x$