RREF Calculator: Row Echelon Form with Steps

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Formula at a glance

Row Echelon Form Calculator: Row operations — Rᵢ ↔ Rⱼ, Rᵢ → kRᵢ, Rᵢ → Rᵢ + kRⱼ. Example: 3 equations, 3 unknowns → Reduced row echelon form (RREF): [1, 0, 0, −8; 0, 1, 0, 1; 0, 0, 1, −2] — How the Row Echelon Form Calculator works: Row operations — Rᵢ ↔ Rⱼ, Rᵢ → kRᵢ, Rᵢ → Rᵢ + kRⱼ

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📘 About the Row Echelon Form

Row reduction simplifies a matrix with three operations — swapping rows, multiplying a row by a non-zero number and adding a multiple of one row to another — without changing the solutions of the equations it represents. Row echelon form (REF) has a staircase of leading entries; reduced row echelon form (RREF) has leading 1s with zeros above and below.

Enter a matrix (with the constants as the last column for a system). The calculator lists every row operation, shows the REF and RREF with exact fractions, the rank and pivot columns, and — for augmented matrices — whether the system has a unique solution, none or infinitely many.

🧮 Formula

Row operations

Rᵢ ↔ Rⱼ, Rᵢ → kRᵢ, Rᵢ → Rᵢ + kRⱼ

RREF

leading 1 in each pivot column, zeros elsewhere in that column

✏️ Worked examples

3 equations, 3 unknowns

You enter
Matrix	1 2 -1 -4; 2 3 -1 -11; -2 0 -3 22
Is the last column the right-hand side?	Yes — solve the system

You get
Reduced row echelon form (RREF)	[1, 0, 0, −8; 0, 1, 0, 1; 0, 0, 1, −2]
Row echelon form (REF)	[1, 2, −1, −4; 0, −1, 1, −3; 0, 0, −1, 2]
Rank	3
Pivot columns	1, 2, 3
Unique solution	x1 = −8, x2 = 1, x3 = −2

RREF [1, 0, 0, −8; 0, 1, 0, 1; 0, 0, 1, −2].

Rank-deficient matrix

You enter
Matrix	1 2 3; 2 4 6; 1 1 1
Is the last column the right-hand side?	No

You get
Reduced row echelon form (RREF)	[1, 0, −1; 0, 1, 2; 0, 0, 0]
Row echelon form (REF)	[1, 2, 3; 0, −1, −2; 0, 0, 0]
Rank	2
Pivot columns	1, 2

RREF [1, 0, −1; 0, 1, 2; 0, 0, 0].

🧭 How to use this calculator

What you enter

MatrixRows separated by ; or new lines, entries by spaces or commas. Fractions like 1/2 are fine.
Is the last column the right-hand side?

What you get

Reduced row echelon form (RREF)
Row echelon form (REF)
Rank
Pivot columns

❓ Frequently asked questions

What is the difference between REF and RREF?

REF only needs zeros below each pivot. RREF also needs each pivot to be 1 with zeros above it, and it is unique.

How does RREF solve a system?

Each row of the RREF of the augmented matrix reads directly as x₁ = …, x₂ = ….

What is the rank of a matrix?

The number of pivots in its echelon form — the number of independent rows (or columns).

≡ Row Echelon Form (RREF) Calculator with Steps