Transpose of Eigenvectors Properties

Question

Given $v$ an eigenvector of an $m \times m$ matrix $A$ with eigenvalue of $5$, where $|v| = 1$, is it the case that $v'*A'*A*v = 5$?

$A*v = 5*v \Rightarrow$

$A'*A*v = 5*A'*v \Rightarrow$

$v'*A'*A*v = 5*v'*A'*v \Rightarrow $

$(A*v)'*(A*v) = 5*v'*A'*v$.

Stuck here

Answer 1

No, $Av = 5v$ and so $(5v)' = (Av)' = v'A'$. Thus

$$v'A'Av = (5v')(5v) = 5^2v'v = 25|v|^2 = 25$$

Answer 2

No:

$$(Av)'(Av) = (5v)'(5v) = 25 v'v = 25 |v|^2 = 25$$