# Definition:Echelon Matrix/Echelon Form

## Definition

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

$\mathbf A$ is in **echelon form** if and only if:

- $(1): \quad$ The leading coefficient in each non-zero row is $1$
- $(2): \quad$ The leading $1$ in any non-zero row occurs to the right of the leading $1$ in any previous row
- $(3): \quad$ The non-zero rows appear before any zero rows.

## Also known as

An **Echelon Matrix** and a matrix in **echelon form** are the same thing.

A matrix in **echelon form** is also sometimes seen as being in **row echelon form**.

It is noted that there appears to be no equivalent definition in the literature for the concept of **column echelon form**, although its structure would be analogous.

## Also defined as

Some sources do not require that, for a matrix to be in **echelon form**, the leading coefficient in each non-zero row must be $1$.

Such a matrix is detailed in Non-Unity Variant of Echelon Matrix.

## 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

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**echelon form** - 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 $1.4$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**echelon form** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**echelon form** - 2021: Richard Earl and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(6th ed.) ... (previous) ... (next):**echelon form**