# Definition:Echelon Matrix/Reduced Echelon Form

## Definition

Let $\mathbf A = \sqbrk a_{m n}$ be a matrix in echelon form whose order is $m \times n$.

The matrix $\mathbf A$ is in **reduced echelon form** if and only if, in addition to being in echelon form, the leading $1$ in any non-zero row is the only non-zero element in the column in which that $1$ occurs.

Such a matrix is called a **reduced echelon matrix**.

## Also known as

The **reduced echelon form** is also known as **row canonical form**, or **reduced row echelon form**.

The abbreviated term **ref** or **rref** is often used for **reduced (row) echelon form**, but it is recommended that it be explained when first invoked in an argument.

## Also see

- Results about
**echelon matrices**can be found here.

## Linguistic Note

An **echelon** is:

*a formation of troops, ships, aircraft, or vehicles in parallel rows with the end of each row projecting further than the one in front.*

It derives from the French word **échelon**, which means a **rung of a ladder**, which describes the shape that this formation has when viewed from above or below.

It is pronounced ** e-shell-on** or something like

**, where the first**

*ay*-shell-on**ay**is properly the French

**é**.

Avoid the pronunciation ** et-chell-on**, which is technically incorrect.

## Sources

- 1982: A.O. Morris:
*Linear Algebra: An Introduction*(2nd ed.) ... (previous) ... (next): Chapter $1$: Linear Equations and Matrices: $1.2$ Elementary Row Operations on Matrices: Definition $1.4$