Prove that if $v$ is an eigenvector, then $A^2v=c^2v$

Question

Prove that if $v$ is an eigenvector for the matrix $A$, then $A^2v=c^2v$

Pretty much all I have is:

$Av=cv$ where $v$ is a nonzero vector

Answer 1

$$Av=cv$$

$$AAv=Acv$$

c is a scalar you can factor it out

$$AAv=c(Av)$$

$$A^2v=c(Av)$$

But you know that $Av=cv$ So

$$A^2v=c(cv)=c^2v$$

Answer 2

Apply $A$ both sides of $Av=cv$ (i.e. $A^2v=Acv$), and use that $A$ commutes with multiplication by constants...