reduced row echelon form

Reduced row echelon form is the fully cleaned-up state of a matrix — the “final answer” shape you steer toward with row operations. Once a matrix reaches it, the solution to the system practically reads itself off the page.

A matrix is in this form when four things hold: any all-zero rows sit at the bottom; the first nonzero entry of every other row is a 1, called a leading 1 (or pivot); each leading 1 sits strictly to the right of the one above it, giving a staircase; and each leading 1 is the only nonzero entry in its entire column. The last condition is what makes it “reduced” rather than just plain echelon form.

Every matrix has exactly one reduced row echelon form, no matter how you do the row operations — it is unique. For a solvable system this final grid shows each variable equal to a constant directly; it also instantly exposes free variables, infinitely many solutions, or an inconsistent system with no solution at all.