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**1. Prove that the rank of a matrix is invariant under similarity.**

**Notes so far:**

Let A, B, P be nxn matrices, and let A and B be similar. That is, there exists an invertible matrix P such that B = P

^{-1}AP. I know the following relations so far: rank(P)=rank(P

^{-1})=n ; rank(A) = rank(A

^{T}); rank(A) + nullity(A) = n . However, I'm unable to write a full proof of the theorem. It makes sense intuitively, but I really would like a written proof.

Thanks for your help!