Row Echelon Form

 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.

Specifically, a matrix is in row echelon form if

• All nonzero rows (rows with at least one nonzero element) are above any rows of all zeroes (all zero rows, if any, belong at the bottom of the matrix).
• The leading coefficient (the first nonzero number from the left, 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.).
• All entries in a column below a leading entry are zeroes (implied by the first two criteria).

Reduced Row Echelon Form

• It is in row echelon form.
• Every leading coefficient is 1 and is the only nonzero entry in its column.

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.

Transformation to Row Echelon Form

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.