Let x and y be eigenvectors of a matrix A such that is also an eigenvector of A. Is an eigenvector of A? Prove or disprove.

I know that and and but I'm not sure where I can go from there without being given that x and y correspond to the same eigenvalue... I started with: but I don't know how to proceed. Please help!!

OP edit after getting help:Thank you--I didn't think of the 2 cases, it becomes easy!