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In linear algebra, a matrix is in echelon form if it has the shape resulting of a Gaussian elimination. Row echelon form means that Gaussian elimination has operated on the rows and column echelon form means that Gaussian elimination has operated on the columns. In other words, a matrix is in column echelon form if its transpose is in row echelon form. Therefore only row echelon forms are considered in the remainder of this article. The similar properties of column echelon form are easily deduced by transposing all the matrices.
Specifically, a matrix is in row echelon form if
This is an example of a 3×5 matrix in row echelon form:
A matrix is in reduced row echelon form (also called row canonical form) if it satisfies the following conditions:
The reduced row echelon form of a matrix may be computed by Gauss–Jordan elimination. Unlike the row echelon form, the reduced row echelon form of a matrix is unique and does not depend on the algorithm used to compute it.
This is an example of a matrix in reduced row echelon form:
Note that this does not always mean that the left of the matrix will be an identity matrix, as this example shows.
For matrices with integer coefficients, the Hermite normal form is a row echelon form that may be computed using Euclidean division and without introducing any rational number nor denominator. On the other hand, the reduced echelon form of a matrix with integer coefficients generally contains noninteger entries.
By means of a finite sequence of elementary row operations, called Gaussian elimination, any matrix can be transformed to row echelon form. Since elementary row operations preserve the row space of the matrix, the row space of the row echelon form is the same as that of the original matrix.
The resulting echelon form is not unique; for example, any multiple by a scalar of a matrix in echelon form is also an echelon form of the same matrix. However, every matrix has a unique reduced row echelon form. This means that the nonzero rows of the reduced row echelon form are the unique reduced row echelon generating set for the row space of the original matrix.
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 equations is said to be in reduced row echelon form or 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 inconsistent, if and only if one of the equations of the canonical form is reduced to 1 = 0. 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.
The Wikibook Linear Algebra has a page on the topic of: Row Reduction and Echelon Forms 
