Proof of Eigenvalue of A^2 When λ is Eigenvalue of A

[sana2476]

Homework Statement

Let λ be an eigenvalue of A. Then λ^2 is an eigenvalue of A^2

The Attempt at a Solution

I know I have to start by using the fact that λ is an e.v of A then set up an equation relating the eigenvalues and vectors to A which is: Ax=λx. And I understand that the equation for λ^2 and A^2 would be: A^2(x)=λ^2(x)...but i don't know how to prove that :-(

 

[kof9595995]

Ax=λx,so AAX=Aλx=λAX=λ*λx,namely A^2(x)=λ^2(x)