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A [[system of linear equations]] is said to be in ''row echelon form'' if its [[augmented matrix]] is in row echelon form. Similarly, a system of linear equations is said to be in ''reduced row echelon form'' or in ''canonical form'' if its augmented matrix is in reduced row echelon form.
The canonical form may be viewed as an explicit solution of the linear system. In fact, the system is [[System of linear equations#Consistency|inconsistent]] if and only if one of the equations of the canonical form is reduced to 0 = 1; that is if there is a leading {{mvar|1}} in the column of the constant terms.<ref>{{Cite book|url=https://books.google.com/books?id=S0imN2tl1qwC|title=Linear Algebra: Theory and Applications|last=Cheney|first=Ward|last2=Kincaid|first2=David R.|date=2010-12-29|publisher=Jones & Bartlett Publishers|isbn=9781449613525|pages=47–50|language=en}}</ref> Otherwise, regrouping in the right hand side all the terms of the equations but the leading ones, expresses the variables corresponding to the pivots as constants or linear functions of the other variables, if any.
== Transformation to row echelon form ==
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