(112013005){\displaystyle {\begin{pmatrix}1&1&2\0&1&3\0&0&5\end{pmatrix}}}
Row swapping. Scalar multiplication. Any row can be replaced by a non-zero scalar multiple of that row. Row addition. A row can be replaced by itself plus a multiple of another row.
(112123345){\displaystyle {\begin{pmatrix}1&1&2\1&2&3\3&4&5\end{pmatrix}}}
For our matrix, the first pivot is simply the top left entry. In general, this will be the case, unless the top left entry is 0. If this is the case, swap rows until the top left entry is non-zero. By their nature, there can only be one pivot per column and per row. When we selected the top left entry as our first pivot, none of the other entries in the pivot’s column or row can become pivots.
For our matrix, we want to obtain 0’s for the entries below the first pivot. Replace the second row with itself minus the first row. Replace the third row with itself minus three times the first row. These row reductions can be succinctly written as R2→R2−R1{\displaystyle R_{2}\to R_{2}-R_{1}} and R3→R3−3R1. {\displaystyle R_{3}\to R_{3}-3R_{1}. } (11201101−1){\displaystyle {\begin{pmatrix}1&1&2\0&1&1\0&1&-1\end{pmatrix}}}
R3→R3−R2{\displaystyle R_{3}\to R_{3}-R_{2}} (11201100−2){\displaystyle {\begin{pmatrix}1&1&2\0&1&1\0&0&-2\end{pmatrix}}} This matrix is in row-echelon form now.
