Reduced Row Echelon Form
Compare to row echelon form, we not only eliminate numbers below the pivot, but also eliminate those above the pivot. We gets reduced row echelon form from Gauss–Jordan elimination.
Example and counter-example
The following matrix is in reduced row echelon form
\left(\begin{array}{llll} 1 & 2 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{array}\right)
\left(\begin{array}{llll} 1 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 1 \end{array}\right)
In reduced row echelon form, all pivots are 1 with 0 above and below. Columns with pivots are called “pivot columns.”