Row echelon form

In linear algebra, a 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 operations. 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.

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.

A matrix is in reduced row echelon form if it is in row echelon form, with the additional property that the first nonzero entry of each row is equal to ${\displaystyle 1}$ and is the only nonzero entry of its column. The reduced row echelon form of a matrix is unique and does not depend on the sequence of elementary row operations used to obtain it. The specific type of Gaussian elimination that transforms a matrix to reduced row echelon form is sometimes called Gauss–Jordan elimination.

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.

A matrix is in row echelon form if

Some texts add the condition that the leading coefficient must be 1 while others require this only in reduced row echelon form.

These two conditions imply that all entries in a column below a leading coefficient are zeros.

The following is an example of a ${\displaystyle 4\times 5}$ matrix in row echelon form, but not in reduced row echelon form: