Definition:Echelon Matrix/Echelon Form

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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.

The definition of column echelon form is directly 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 ay-shell-on, where the first ay is properly the French é.

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