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row echelon form

A staircase shape: leading entries march right going down, zero rows at the bottom.

An augmented matrix is in row echelon form if any all-zero rows sit at the bottom, each nonzero row's leading entry (its pivot) sits strictly right of the leading entry above it, and every entry directly below a leading entry is 0. Gaussian elimination is the column-by-column procedure that walks any matrix to this form, after which back-substitution solves the last row first and climbs the staircase upward.

See also

Gaussian elimination pivot reduced row echelon form
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