To record row reduction of $A\in\mathbb F^{m\times n}$, augment by $I_m$ and perform operations giving $[A\ I_m]\to[R\ E]$, where $R=\operatorname{RREF}(A)$ and $EA=R$. Here $E$ is invertible and records the row operations; it need not be unique for rank-deficient $A$.
One may reduce the left block while carrying the right block along. Do not assume that an arbitrary stage reducing only $A$ has fully reduced the entire augmented array. If $A$ is square invertible, $R=I$ and $E=A^{-1}$.
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