Definition RREF Reduced Row-Echelon Form

A matrix is in reduced row-echelon form if it meets all of the following conditions:

1. If there is a row where every entry is zero, then this row lies below any other row that contains a nonzero entry.
2. The leftmost nonzero entry of a row is equal to 1.
3. The leftmost nonzero entry of a row is the only nonzero entry in its column.
4. Consider any two different leftmost nonzero entries, one located in row $i$, column $j$ and the other located in row $s$, column $t$. If $s>i$, then $t>j$.

A row of only zero entries is called a zero row and the leftmost nonzero entry of a nonzero row is a leading 1. A column containing a leading 1 will be called a pivot column. The number of nonzero rows will be denoted by $r$, which is also equal to the number of leading 1's and the number of pivot columns.

The set of column indices for the pivot columns will be denoted by $D=\set{d_1,\,d_2,\,d_3,\,\ldots,\,d_r}$ where $d_1<d_2<d_3<\cdots<d_r$, while the columns that are not pivot columns will be denoted as $F=\set{f_1,\,f_2,\,f_3,\,\ldots,\,f_{n-r}}$ where $f_1<f_2<f_3<\cdots<f_{n-r}$.