row echelon form

Row echelon form

A matrix is in row echelon form if it can be obtained as a result of Gaussian elimination.
There are two requirements for a matrix to be in row echelon form:

All entries in the column below must be 0
All rows that contain zeroes are at the bottom of the matrix.

Example:

\begin{align*} \begin{pmatrix} \ast & \ast & \ast & \ast & \ast \\ 0 & \ast & \ast & \ast & \ast \\ 0 & 0 & 0 & \ast & \ast \\ 0 & 0 & 0 & 0 & 0 \\ \end{pmatrix} \end{align*}

where the first \[\ast\] of each row represents a nonzero number and the consecutive \[\ast\] representing any number.

Reduced row echelon form

For a system of two linear equations, the goal of Gaussian elimination is to convert the part of the augmented matrix left of the dividing line into the identity matrix. For many systems, it is not possible to reach the identity in the augmented matrix via Gaussian elimination. In any case, a certain version of the matrix that has the maximum number of components eliminated is said to be the Reduced row echelon form.
The additional two requirements (on top of the original two) would be that:

The pivot (leading entry) of every nonzero row is one
Every pivot must be the only nonzero element in that column.

Example:

\begin{align*} \begin{pmatrix} 1 & 0 & \ast & 0 & \ast \\ 0 & 1 & \ast & 0 & \ast \\ 0 & 0 & 0 & 1 & \ast \\ 0 & 0 & 0 & 0 & 0 \\ \end{pmatrix} \end{align*}

Referenced by:

Gaussian elimination