row echelon form

Row echelon form is the staircase shape a matrix settles into after elimination. The first nonzero entry in each row, called a pivot or leading entry, sits to the right of the pivot in the row above, and every entry below a pivot is zero. Any all-zero rows sink to the bottom.

Picture a descending staircase of leading entries stepping rightward as you read down the rows — that visual is exactly what 'echelon' means. From this shape the structure of the system is plain to see.

Reduced row echelon form (RREF) goes one step further: each pivot is made equal to 1, and every entry directly above a pivot is also cleared to zero. RREF is unique for a given matrix and reads out the solution almost without any extra arithmetic.