RREF Calculator — Reduced Row Echelon Form by Gauss-Jordan Elimination

How it works

Row operations: swap Rᵢ ↔ Rⱼ · scale Rᵢ → Rᵢ ÷ pivot · eliminate Rₖ → Rₖ − c·Rᵢ · RREF: each pivot is 1, the only non-zero entry in its column, and pivots move right row by row

Gauss-Jordan elimination uses the three elementary row operations to turn a matrix into reduced row echelon form (RREF), which is unique for every matrix. Working column by column, it picks a non-zero pivot, scales its row so the pivot is 1, and clears every other entry in that column. The number of pivots is the rank. For an augmented matrix [A | b], a pivot in the last column means the system is inconsistent; otherwise columns without a pivot are free variables.

Worked example

For the system 2x₁ + x₂ − x₃ = 8, −3x₁ − x₂ + 2x₃ = −11, −2x₁ + x₂ + 2x₃ = −3, row reduction of the augmented matrix gives [1, 0, 0 | 2] [0, 1, 0 | 3] [0, 0, 1 | −1]: rank 3, pivots in columns 1, 2, 3, and the unique solution x₁ = 2, x₂ = 3, x₃ = −1.

Sources: Gaussian elimination — Wikipedia · Row Echelon Form — Wolfram MathWorld

Frequently asked questions