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In [[linear algebra]], a [[Matrix (mathematics)|matrix]] is in '''row echelon form''' if it can be obtained as the result of [[Gaussian elimination]]. Every matrix can be put in row echelon form by applying a sequence of [[elementary row operation]]s. The term ''echelon'' comes from the French ''échelon'' ("level" or step of a ladder), and refers to the fact that the nonzero entries of a matrix in row echelon form look like an inverted staircase.
[[File:Row echelon form.png|thumb|right|Example of a rectangular matrix in row echelon form]]
For [[square matrices]], an [[upper triangular matrix]] with nonzero entries on the diagonal is in row echelon form, and a matrix in row echelon form is (weakly) upper triangular. Thus, the row echelon form can be viewed as a generalization of upper triangular form for rectangular matrices.
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A matrix is in '''column echelon form''' if its [[transpose]] is in row echelon form. Since all properties of column echelon forms can therefore immediately be deduced from the corresponding properties of row echelon forms, ''only row echelon forms are considered in the remainder of the article.''
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A matrix is in '''row echelon form''' if
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