🧮 RREF Calculator (Matrix)

Enter the coefficients of a system of three linear equations in three unknowns (x, y, z). The calculator reduces the associated matrix step by step to reduced row echelon form (RREF) using Gauss-Jordan elimination and reads off the solution.

Row 1: x coefficient

Row 1: y coefficient

Row 1: z coefficient

Row 1: constant

Row 2: x coefficient

Row 2: y coefficient

Row 2: z coefficient

Row 2: constant

Row 3: x coefficient

Row 3: y coefficient

Row 3: z coefficient

Row 3: constant

Solution: x 5

Solution: y 3

Solution: z -2

Formula

RREF via Gauss-Jordan elimination: every pivot becomes 1, every other value in that column becomes 0

The calculator turns the system into an augmented matrix (the coefficients plus the constants) and works column by column: it picks a pivot row, scales it so the pivot becomes 1, and subtracts multiples of that row from every other row until the rest of the column is 0. If the result gives the identity matrix in the first three columns, the solution sits directly in the last column; otherwise the system is dependent or inconsistent.

Steps

1Enter the coefficients and constants of each of the three equations.
2The calculator performs Gauss-Jordan elimination until the matrix is in RREF.
3Read x, y and z directly if the last column contains a unique solution.

Allowed row operations in Gauss-Jordan elimination

Operation	Example	Effect
Swap rows	R1 ↔ R2	Changes the order, not the solution
Scale a row	R1 → R1 ÷ 2	Makes the pivot equal to 1
Add/subtract a row	R2 → R2 − 3×R1	Makes other values in the column 0

These three operations never change the solution of the system, only how the matrix looks.

limitations

This calculator only supports systems of exactly three equations in three unknowns.

commonMistakes

Forgetting to enter 0 as the coefficient for a variable that doesn't appear in an equation.
Assuming a non-identity matrix always means an error, when it can simply be a dependent or inconsistent system.
Mixing up the row order when manually checking an intermediate step.

useCases

Solving a system of three equations in three unknowns for linear algebra coursework.
Checking homework or an exam without writing out the intermediate steps by hand.
Quickly checking whether a system has a unique solution, infinitely many, or none.