Matrix Algebra Questions

Question

I would like you to help me with two questions I am stuck in. You can call these homework questions. It would be helpful if you can give me non-trivial hints instead of complete solution.

1) Let $A$ be a matrix such that $Rowspace(A)=Coulmnspace(A)$. Prove or disprove that A is symmetric.

2) Suppose $A$ and $B$ are two matrices such that $Columnspace(A)=Columnspace(B)$ and $BA=AB$. Prove or disprove that $A=B$

Answer 1

Counterexample for both:

$$ A = \pmatrix{0&1&0\\0&0&1\\1&0&0}, \quad B = \pmatrix{1&0&0\\0&1&0\\0&0&1} $$

Answer 2

1. Two vectors can be composed from the same basis, but not be equal.
2. Did you mis-transcribe the question? Equality is symmetric for matrices. BA $=$ AB $\implies$ AB $=$ BA