In the following matrix equation use elementary operation $R_{2} \rightarrow R_{2}+R_{1}$ and the equation thus obtained:
$\left[\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right]\left[\begin{array}{rr}1 & 0 \\ 2 & -1\end{array}\right]=\left[\begin{array}{ll}8 & -3 \\ 9 & -4\end{array}\right]$
$\left[\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right]\left[\begin{array}{rr}1 & 0 \\ 2 & -1\end{array}\right]=\left[\begin{array}{rr}8 & -3 \\ 9 & -4\end{array}\right]$
By applying elementary operation $R_{2} \rightarrow R_{2}+R_{1}$, we get
$\left[\begin{array}{ll}2 & 3 \\ 3 & 7\end{array}\right]\left[\begin{array}{rr}1 & 0 \\ 2 & -1\end{array}\right]=\left[\begin{array}{cc}8 & -3 \\ 17 & -7\end{array}\right]$ (Every row operation is equivalent to left-multiplication by an elementary matrix.)
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