This is from linear algebra Prove or give a countexample A I

This is from linear algebra:

Prove or give a countexample

A) If B is similar to A and A is symmetric, then B is symmetric

B) If B is similar to A, then N(B)=N(A)

C) If B is similar to A, then rank(B)=rank(A)

Solution

A) if A is symmetric and B is similar to A that B must also be symmetric. For example consider A to be the 2x2 matrix with first row 0, 1 and second row 1, 0. Let S denote the 2x2 matrix with first row 1, 0 and second row 1, 1. You can check that S is invertible and that S^(-1) A S is the matrix with first row 1, 1 and second row 0, -1. Now A is similar to S^(-1) A S by definition, and A is symmetric, but S^(-1) A S is not.

C)

Let A=PBQ where P,Q are invertible.

Rank(A)=Rank(PBQ)Rank(PB)Rank(B)=Rank(P1AQ1)Rank(A) so all inequalities must be equalities.

So Rank(A)=Rank(B).