RREF

Definition

Reduced row echelon form is defined as a row echelon form with the following additional property:

• Each column that contains a leading 1 has zeros everywhere else in that column.

Remark

For example,

$$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$

Solution set of a linear system can be obtained by its reduced row echelon form. For example, consider the following augmented matrix:

$$\left[\begin{array}{ccccc}1& 0& 0& 0& 3\\ 0& 1& 0& 0& -1\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 5\end{array}\right]$$

This matrix is in reduced row echelon form and correspond the system of equation

$$\begin{array}{cccccc}{x}_1& & & & =& 3\\ & x_2& & & =& -1\\ & & x_3& & =& 0\\ & & & x_4& =& 5\end{array}$$

Thus, the system has 1 solutions, which is $(3,-1,0,5)$.

Tip

We can use elementary row operations to transform an augmented matrix to a reduced row echelon form to obtain its solution.