Example EMRO Elementary matrices and row operations

We will perform a sequence of row operations (Definition RO) on the $3\times 4$ matrix $A$, while also multiplying the matrix on the left by the appropriate $3\times 3$ elementary matrix. \begin{equation*} A= \begin{bmatrix} 2 & 1 & 3 & 1\\ 1 & 3 & 2 & 4\\ 5 & 0 & 3 & 1 \end{bmatrix} \end{equation*} \begin{align*} \rowopswap{1}{3}:\ & \begin{bmatrix} 5 & 0 & 3 & 1\\ 1 & 3 & 2 & 4\\ 2 & 1 & 3 & 1 \end{bmatrix} & \elemswap{1}{3}:\ & \begin{bmatrix} 0 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 0 \end{bmatrix} \begin{bmatrix} 2 & 1 & 3 & 1\\ 1 & 3 & 2 & 4\\ 5 & 0 & 3 & 1 \end{bmatrix} = \begin{bmatrix} 5 & 0 & 3 & 1\\ 1 & 3 & 2 & 4\\ 2 & 1 & 3 & 1 \end{bmatrix}\\ \rowopmult{2}{2}:\ & \begin{bmatrix} 5 & 0 & 3 & 1\\ 2 & 6 & 4 & 8\\ 2 & 1 & 3 & 1 \end{bmatrix} & \elemmult{2}{2}:\ & \begin{bmatrix} 1 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 5 & 0 & 3 & 1\\ 1 & 3 & 2 & 4\\ 2 & 1 & 3 & 1 \end{bmatrix} = \begin{bmatrix} 5 & 0 & 3 & 1\\ 2 & 6 & 4 & 8\\ 2 & 1 & 3 & 1 \end{bmatrix}\\ \rowopadd{2}{3}{1}:\ & \begin{bmatrix} 9 & 2 & 9 & 3\\ 2 & 6 & 4 & 8\\ 2 & 1 & 3 & 1 \end{bmatrix} & \elemadd{2}{3}{1}:\ & \begin{bmatrix} 1 & 0 & 2\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 5 & 0 & 3 & 1\\ 2 & 6 & 4 & 8\\ 2 & 1 & 3 & 1 \end{bmatrix} = \begin{bmatrix} 9 & 2 & 9 & 3\\ 2 & 6 & 4 & 8\\ 2 & 1 & 3 & 1 \end{bmatrix} \end{align*}