Prove or give a counterexample:
If b is similar to A, then rank(B)=rank(A).
The Attempt at a Solution
This is obvious when A and B have maximum rank (take the determinant of both sides of the similarity relation). My intuition and all the examples I have looked at tell me it is also true when they have less than maximum rank. But how to prove it? Is it easier to look at the nullity and use the Nullity-Rank Theorem?