Row echelon form

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In mathematics, a matrix is in row echelon form if is satisfies the following requirements:

All nonzero rows are above any rows of all zeroes.
The leading coefficient of a row is always strictly to the right of the leading coefficient of the row above it.

Furthermore, in some definitions:

The leading coefficient of each nonzero row has to be one.^[1]

Row echelon form is closely related to reduced row echelon form (row canonical form). The difference is that in reduced row echelon form, the entries above the leading coefficient also have to be zero, and that the leading coefficient is always one.

The first non-zero entry in each row is called a pivot.

1 Examples
2 Non-uniqueness
3 Systems of linear equations
4 See also
5 Notes

[edit] Examples

This matrix is in row echelon form:

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

The following matrix is also in row echelon form, unless the leading coefficient of each nonzero row is required to be one:

$\begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 9 & 0 & 2 \\ 0 & 0 & 0 & 3 \\ \end{bmatrix}$

However, this matrix is not in row echelon form, as the leading coefficient of row 3 is not strictly to the right of the leading coefficient of row 2.

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

[edit] Non-uniqueness

Every non-zero matrix can be reduced to an infinite number of echelon forms (they can all be multiples of each other, for example) via elementary matrix transformations. However, all matrices and their row echelon forms correspond to exactly one matrix in reduced row echelon form.

[edit] Systems of linear equations

A system of linear equations is said to be in echelon form if its augmented matrix is in row echelon form. Similarly, a system of equations is said to be in reduced echelon form or canonical form if its augmeneted matrix is in reduced row echelon form.