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 $A=B$