A superspace of an eigenspace

Question

Let $a$ be an eigenvalue of a matrix $A$, then the eigenspace is given by $E_a=\{v\in V\mid(A-aI)v=0\}$.

Now define $F_a=\{v\in V\mid (A-aI)^tv=0$ for some integer $t\geq 1\}$. Clearly $F_a$ is a subspace of $V$ containing $E_a$, $E_a\subset F_a$, but I'm not sure if $F_a\subset E_a$?

I wish to show if $t$ is the smallest integer such that $(A-aI)^tv=0$, then $t=1$.

Answer 1

try $$ \left( \begin{array}{cc} 5 & 1 \\ 0 & 5 \end{array} \right) $$

Then $$ \left( \begin{array}{ccc} 7 & 1 &0 \\ 0 & 7 & 1 \\ 0 & 0 & 7 \end{array} \right) $$