# All matrices that share a null row space are obtainable from one another by elmentary row operations?

Given an $m\times n$ matrix $\mathbb{A}$, the set of $n$-vectors $\mathbf{x}$ that satisfy $\mathbb{A}\cdot\mathbf{x}=0$ is the null row-space of $\mathbb{A}$.

The elementary row operations on $\mathbb{A}$ are: summing one row to another row and multiply a row by a nonzero constant.

The question is: For all $m\times n$ matrices $\mathbb{B}$ that have the same null row-space as $\mathbb{A}$ (that is, $\mathbb{A}\cdot\mathbf{x}=0$ iff $\mathbb{B}\cdot\mathbf{x}=0$), can $\mathbb{B}$ be obtained from $\mathbb{A}$ by elementary row operations on $\mathbb{A}$?

Hint As the row operations can be reversed, it is enough to prove/disprove the result in the case $A,B$ are Reduced Row Echelon Forms (RREF).
Therefore, the problem reduces to: If $A,B$ are RREF, is it true that they have the same Null space if and only if they are equal?