reduced row echelon form

Row echelon form with every pivot equal to 1 and cleared above as well as below.

A matrix is in reduced row echelon form (RREF) if it is in row echelon form and, further, every pivot equals $1$ and every pivot is the only nonzero entry in its column (cleared above, not just below). Gauss-Jordan elimination reaches RREF with a forward elimination pass followed by a backward cleanup pass. Unlike row echelon form, RREF is unique: every legal sequence of row operations on the same matrix converges to the same RREF.