Reduced Row Echelon Form Calculator (RREF) — Gauss-Jordan & Gauss Elimination with Steps

Welcome to the reduced row echelon form calculator (or rref calculator for short), where we'll solve a system of equations of your choice using matrix row reduction and elementary row operations. We also give you the option to choose whether you'd like the reduced version or not. Based on the choice you make, our tool can be viewed as a Gauss-Jordan elimination calculator (with the first variant) or a Gauss elimination calculator. Moreover, in case your system has infinitely many solutions, our rref calculator will even tell you what they look like!

What is a system of equations?

Remember all those math scenarios that try to imitate real life? Whenever we have some value that we don't know but which must satisfy some property, we describe that connection using an equation. We denote the unknown value with a symbol called a variable, and we write what we know about it using operations such as addition, subtraction, multiplication, or division.

If we have several equations and want all of them to be satisfied by the same numbers, then we're dealing with a system of equations. Usually there is more than one variable in total, and the most common problems have the same number of equations as variables. For example, from a real-life riddle we might get the system:

x + y + z = 32
-x + 2y = 25
-y + 2z = 16

Once written this way, the whole system can be captured by a rectangular grid of numbers — a matrix — which is exactly what the rref calculator works on.

Elementary row operations

To solve the system we place its coefficients (and, for an augmented matrix, the right-hand sides) into a matrix and simplify it using three elementary row operations. They never change the solution set:

1. Row swap — interchange two rows: Ri ↔ Rj.
2. Scaling — multiply a row by a non-zero number: Ri → k·Ri.
3. Row addition — add a multiple of one row to another: Ri → Ri + k·Rj.

Applying these operations step by step is called matrix row reduction. Our calculator prints every operation it performs, together with the matrix after each stage, so you can follow the whole reduction.

Gauss-Jordan elimination vs Gauss elimination

Row reduction can stop at two different "shapes":

• Row echelon form (REF) — produced by Gauss elimination. Every leading (pivot) entry is a 1, each pivot sits to the right of the one above it, and all entries below each pivot are zero. The rows below the last pivot are all zeros. From REF you finish with back-substitution.
• Reduced row echelon form (RREF) — produced by Gauss-Jordan elimination. In addition to the REF conditions, every entry above each pivot is also zero, so each pivot column contains a single 1. In RREF the solution can simply be read off — no back-substitution needed.

Toggle the Reduction selector to switch our tool between a Gauss-Jordan (RREF) calculator and a Gauss (REF) calculator. Either way you also get the matrix rank and the list of pivot columns.

Example: using the reduced row echelon form calculator

Let's solve the 3×3 system:

2x + 3y + z = 9
x - y + z = 0
x + 2y - z = 4

1. Set Rows to 3 and Columns to 4 (three coefficient columns plus the constants column).
2. Tick Treat as augmented matrix so the last column is read as the right-hand side.
3. Keep Reduced row echelon (Gauss-Jordan) selected.
4. Enter the rows: 2 3 1 9, 1 -1 1 0, and 1 2 -1 4.
5. Read the result: the left block becomes the identity matrix and the calculator reports x = 1, y = 2, z = 1, together with every elementary row operation used to get there.

Metric and US units

Row reduction is pure algebra, so units never change the arithmetic. When you interpret an augmented matrix as a real-world system, use the unit selector to label the solved values — either metric (meters, kilograms, liters) or US customary (feet, pounds, gallons) — so the answer reads in the units your problem is stated in.

Frequently Asked Questions