Row Echelon Form

Overview

Row echelon form (REF) and reduced row echelon form (RREF) are standardized matrix forms that make it easy to identify the solutions of a linear system.

Row Echelon Form (REF)

A matrix is in row echelon form if:

1. All zero rows are at the bottom
2. The leading entry (pivot) of each non-zero row is to the right of the pivot above it
3. All entries below a pivot are zero

Example of REF

$\begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 5 & 6 & 7 \\ 0 & 0 & 8 & 9 \\ 0 & 0 & 0 & 0 \end{bmatrix}$

Reduced Row Echelon Form (RREF)

A matrix is in RREF if it is in REF and additionally:

4. Each pivot is 1
5. Each pivot is the only non-zero entry in its column

Example of RREF

$\begin{bmatrix} 1 & 0 & 0 & a \\ 0 & 1 & 0 & b \\ 0 & 0 & 1 & c \\ 0 & 0 & 0 & 0 \end{bmatrix}$

Pivot and Free Variables

Pivot Columns

Columns containing a leading 1 (pivot position).

Variables corresponding to pivot columns are called basic variables.

Free Columns

Columns without a pivot.

Variables corresponding to free columns are called free variables.

Identifying Pivots

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

Pivots are in columns 1 and 3.

• Basic variables: $x_1$, $x_3$
• Free variables: $x_2$, $x_4$

Reading Solutions from RREF

Unique Solution

$\left[\begin{array}{ccc|c} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 4 \end{array}\right]$

Solution: $x = 2$, $y = 3$, $z = 4$

Infinitely Many Solutions

$\left[\begin{array}{ccc|c} 1 & 2 & 0 & 3 \\ 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 \end{array}\right]$

$x_2$ is free. Let $x_2 = t$.

$x_1 = 3 - 2t$, $x_3 = 4$

Solution: $(3-2t, t, 4)$ for any $t \in \mathbb{R}$

No Solution

$\left[\begin{array}{ccc|c} 1 & 2 & 0 & 3 \\ 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 5 \end{array}\right]$

Last row: $0 = 5$ (contradiction)

No solution exists.

Gauss-Jordan Elimination

Algorithm to reach RREF from any matrix:

1. Perform Gaussian elimination to reach REF
2. Starting from the rightmost pivot:
   • Scale row to make pivot = 1
   • Eliminate all entries above the pivot
3. Move left and repeat

Example: Complete Reduction

$\begin{bmatrix} 2 & 4 & -2 \\ 1 & 3 & 2 \end{bmatrix} \to \begin{bmatrix} 1 & 2 & -1 \\ 1 & 3 & 2 \end{bmatrix} \to \begin{bmatrix} 1 & 2 & -1 \\ 0 & 1 & 3 \end{bmatrix} \to \begin{bmatrix} 1 & 0 & -7 \\ 0 & 1 & 3 \end{bmatrix}$

Steps:

1. $R_1/2 \to R_1$: first row
2. $R_2 - R_1 \to R_2$: create zero below pivot
3. $R_1 - 2R_2 \to R_1$: create zero above pivot

Properties of RREF

Property	Description
Uniqueness	Every matrix has a unique RREF
Rank	Number of pivots = rank
Nullity	Number of free columns = nullity
Invertibility	Square matrix with $I$ as RREF is invertible

Applications

• Solving systems of equations
• Finding matrix rank
• Computing null space basis
• Determining linear independence
• Finding matrix inverse