Unitary equivalence question

[coolmemin]

I need a way to prove that two matrices are unitarily similar because I'm having a hard time doing it with the definition. Here's the question. If A and B are 2x2 similar matrices, and |A|^2=|B|^2, (|| is the norm, I can't write the sum notation thingy's), then A and B are unitarily similar.
I assume since S^-1AS=B that you have to show that S=S^-1 and then you're done but I've been stumped over here. I would appreciate any hints anyone has to this problem

thanks