Row Echelon Form (REF)

Definition

Row echelon form is defined as the augmented matrix of a linear system with the following properties:

1. If a row does not consist entirely of zeros, then the first nonzero number in the row is a 1. We call this a leading 1.
2. If there are any rows that consist entirely of zeros, then they are grouped together at the bottom of the matrix.
3. In any two successive rows that do not consist entirely of zeros, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row.

Remark

For example,

$$\left[\begin{array}{cccc}1& 0& 3& -1\\ 0& 1& -4& 2\end{array}\right]$$

The above augmented matrix correspond to the linear system

$$\begin{array}{ccccc}x& & +3z& =& -1\\ & y& -4z& =& 2\end{array}$$

Since $x$ and $y$ correspond to the leading 1’s in the augmented matrix, we call them leading variables. The remaining variables (in this case $z$) are called free variables.