Reduced Echelon Forms

Row Reduction Algorithm. A pivot position refers to the leftmost nonzero entry of a nonzero row. By using pivot positions and basic row operations, the following procedure produces a reduced echelon form (REF).

Introduce zeros below each pivot position by row operations. Work downward and to the right to reach an echelon form.
Make the values of pivot position to be ones by row operations.
Introduce zeros above the each pivot position by row operations. Work upward and to the left to yield a reduced echelon form.

The size

of pivot positions used in the above procedure is called the rank of

, denoted by $\mathrm{rank}~A$ .

Echelon and Reduced Echelon Form. The following matrix is an example of echelon form. The leading entries ( $\bullet$ ) must be nonzero values, while the starred entries ( $*$ ) may be any value (including zero).

$\displaystyle \left[\begin{array}{rrrrr} \bullet & * & * & * \\ 0 & \bullet & * & * \\ 0 & 0 & 0 & 0 \end{array}\right]$

Next the echelon form can be further reduced to the following reduced echelon form.

$\displaystyle \left[\begin{array}{rrrrr} 1 & 0 & * & * \\ 0 & 1 & * & * \\ 0 & 0 & 0 & 0 \end{array}\right]$

Here the leading entries (pivot entries) are changed to 1's and all the entries below and above each pivot entries are 0's.