Let $A$ and $B$ be a $m \times n$ matrix, $b$ and $c$ be a column vector in $R^m$.
Is it always true that if $(A | b)$ and $(B | c)$ are row equivalent, then $A^TAx=A^Tb$--> (Eqn1) and $B^TBx=B^Tc $-->(Eqn2) have the same solution?
My solution is as such:
$A= E_kE_{k-1}...E_2E_1B = QB$ then $A^T = B^TQ^T$
$A^TAx=B^TQ^TQBx=B^TQ^Tb \neq B^TBx=B^Tc $
So I came to a conclusion that the statement is false. Is this the right approach?