Reduced row echelon form
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In mathematics, a matrix is in reduced row echelon form (also known as row canonical form) if it satisfies the following requirements:
- All nonzero rows are above any rows of all zeroes.
- The leading coefficient of a row is always to the right of the leading coefficient of the row above it.
- All leading coefficients are 1.
- All leading coefficients are the only nonzero entries in a given column (equivalently: all leading coefficients have zeros both above and below them).
It is readily seen that these conditions are stronger than those for row echelon form. Therefore, every matrix in reduced row echelon form is in row echelon form.
Unlike row echelon form, every matrix reduces to a unique matrix in reduced row echelon form by elementary row operations. (See Elementary matrix transformations.)
[edit] Examples
The following matrix is in reduced row echelon form:
However, the following matrix is not in reduced row echelon form, as the 1 in the third row is not the only nonzero entry in its column:
