17.3 Reduced Row Echelon Form

Types: N/A
Examples: N/A
Constructions: N/A
Generalizations: 17.2 Row Echelon Form

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Reduced Row Echelon Form

Let $A \in R^{m \times n}$ be a given matrix. We say $A$ is in reduced row echelon form iff $A$ satisfies the following three conditions:

$A$ is in 17.2 Row Echelon Form
All leading entries of the rows of $A$ are equal to 1
For any column that includes a leading entry, all other coefficients in that column are zero. In other words, every entry underneath and above a leading entry is zero.

Remark. An oversimplified way to think of reduced row echelon form is is to turn the stars into 1s and turn every other value in the corresponding column to 0s.
$⧫$

Let's take a look again at the examples of matrices in 17.2 Row Echelon Form and compare them with their reduced row echelon form.

Example

$[\begin{matrix} ⋆ & \times & \times \\ 0 & ⋆ & \times \\ 0 & 0 & ⋆ \end{matrix}]$

In reduced row echelon form:

[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]

Example

$[\begin{matrix} ⋆ & \times & \times & \times & \times & \times & \times \\ 0 & 0 & ⋆ & \times & \times & \times & \times \\ 0 & 0 & 0 & 0 & 0 & ⋆ & \times \\ 0 & 0 & 0 & 0 & 0 & 0 & ⋆ \end{matrix}]$

In reduced row echelon form:

[\begin{matrix} 1 & \times & 0 & \times & \times & 0 & 0 \\ 0 & 0 & 1 & \times & \times & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}]

Example

$[\begin{matrix} 0 & ⋆ & \times & \times & \times & \times & \times & \times & \times & \times & \times \\ 0 & 0 & ⋆ & \times & \times & \times & \times & \times & \times & \times & \times \\ 0 & 0 & 0 & 0 & ⋆ & \times & \times & \times & \times & \times & \times \\ 0 & 0 & 0 & 0 & 0 & ⋆ & \times & \times & \times & \times & \times \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋆ & \times & \times \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]$

In reduced row echelon form:

[\begin{matrix} 0 & 1 & 0 & \times & 0 & 0 & \times & \times & 0 & \times & \times \\ 0 & 0 & 1 & \times & 0 & 0 & \times & \times & 0 & \times & \times \\ 0 & 0 & 0 & 0 & 1 & 0 & \times & \times & 0 & \times & \times \\ 0 & 0 & 0 & 0 & 0 & 1 & \times & \times & 0 & \times & \times \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \times & \times \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]