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In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination.
A matrix being in row echelon form means that Gaussian elimination has operated on the rows, andcolumn 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
All rows consisting of only zeroes are at the bottom.The leading coefficient (also called the pivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it.Some texts add the condition that the leading coefficient must be 1.
These two conditions imply that all entries in a column below a leading coefficient are zeros.
The following is an example of a 3×5 matrix in row echelon form, which is not in reduced row echelon form (see below):
[ 1 a 0 a 1 a 2 a 3 0 0 2 a 4 a 5 0 0 0 1 a 6 ] {\displaystyle \left[{\begin{array}{ccccc}1&a_{0}&a_{1}&a_{2}&a_{3}\\0&0&2&a_{4}&a_{5}\\0&0&0&1&a_{6}\end{array}}\right]}Many properties of matrices may be easily deduced from their row echelon form, such as the rank and the kernel.
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