$\begin{array}{l}\,\,\,\qquad \qquad\left[\begin{array}{cccc}3&1&-2&4\\5&3&4&2\\4&2&1&3\\6&4&7&5\end{array}\right]\\ \xrightarrow{\,\,r_4 \to r_4 - 2r_1\,}\left[\begin{array}{cccc}3&1&-2&4\\5&3&4&2\\4&2&1&3\\0&2&11&-3\end{array}\right]\\ \xrightarrow{\quad\begin{array}{l}r_1 \to 5r_1\\r_2 \to 3r_2\end{array}\,\,}\left[\begin{array}{cccc}15&5&-10&20\\15&9&12&6\\4&2&1&3\\0&2&11&-3\end{array}\right]\\ \xrightarrow{\,\,\begin{array}{l}r_2 \to r_2 - r_1\\r_1\to \tfrac15r_1\end{array}\,\,}\left[\begin{array}{cccc}3&1&-2&4\\0&4&22&-14\\4&2&1&3\\0&2&11&-3\end{array}\right]\\ \xrightarrow{\quad\begin{array}{l}r_1 \to 4r_1\\r_3 \to 3r_3\end{array}\,\,}\left[\begin{array}{cccc}12&4&-8&16\\0&4&22&-14\\12&6&3&9\\0&2&11&-3\end{array}\right]\\ \xrightarrow{\,\,\begin{array}{l}r_3 \to r_3 - r_1\\r_1 \to \tfrac14r_1\end{array}\,\,}\left[\begin{array}{cccc}3&1&-2&4\\0&4&22&-14\\0&2&11&-7\\0&2&11&-3\end{array}\right]\\ \xrightarrow{\begin{array}{l}r_3 \to r_3 - \tfrac12r_2\\r_4 \to r_4 -\tfrac12r_2\end{array}}\left[\begin{array}{cccc}3&1&-2&4\\0&4&22&-14\\0&0&0&0\\0&0&0&4\end{array}\right]\\ \xrightarrow{\quad r_3 \leftrightarrow r_4\quad}\left[\begin{array}{cccc}3&1&-2&4\\0&4&22&-14\\0&0&0&4\\0&0&0&0\end{array}\right]\\ \end{array}$
