\begin{definition}[Definition:Elementary Operation/Row]
Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over a field $K$.
The '''elementary row operations''' on $\mathbf A$ are operations which act upon the rows of $\mathbf A$ as follows.
For some $i, j \in \closedint 1 m: i \ne j$:
{{begin-axiom}}
{{axiom | n = \text {ERO} 1
        | t = For some $\lambda \in K_{\ne 0}$, multiply row $i$ by $\lambda$
        | m = r_i \to \lambda r_i
}}
{{axiom | n = \text {ERO} 2
        | t = For some $\lambda \in K$, add $\lambda$ times row $j$ to row $i$
        | m = r_i \to r_i + \lambda r_j
}}
{{axiom | n = \text {ERO} 3
        | t = Exchange rows $i$ and $j$
        | m = r_i \leftrightarrow r_j
}}
{{end-axiom}}
\end{definition}