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 to reduced row echelon form (RREF) using Gauss-Jordan elimination and shows the solution.
Result
5
Solution: x
- 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.
Worked examples
x+y+z=6, 2y+5z=โ4, 2x+5yโz=27
โ x=5, y=3, z=โ2.
Common mistakes
- Forgetting to enter 0 as the coefficient for a variable that doesnโt appear in an equation.
Limitations
- This calculator only supports systems of exactly three equations in three unknowns.
When to use it
- Solving a system of three equations in three unknowns for linear algebra coursework.
Frequently asked questions
What does it mean if there is no unique solution?
It means the system contains dependent equations (infinitely many solutions) or is inconsistent (no solution at all). The RREF matrix shown indicates which case applies.
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