How does one prove that if 2 systems of linear equations $Ax=c_1$ and $Bx=c_2$ have the same solution set, then they are row equivalent?
I see that it's true from wikipedia:
[...] two linear systems are equivalent if and only if they have the same solution set.
Original question before edits:
For fixed $m$ and $n$, is it possible to have two $m$ by $n$ matrices $A$ and $B$ that are not row equivalent but have the same solution set? What about if $A$ and $B$ are allowed to be of differing dimensions?