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.
結果
5
Solution: x
- Solution: y
- 3
- Solution: z
- -2
公式
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.
ステップ解説
- 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.
計算例
x+y+z=6, 2y+5z=−4, 2x+5y−z=27
→ x=5, y=3, z=−2.
x+y=2, 2x+2y=4, x−y=0
→ No unique solution: the second equation is a multiple of the first, the system is dependent.
x+y+z=3, x+y+z=5, x−y=1
→ No solution: the first two equations contradict each other, the system is inconsistent.
Reference table
| 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.
よくある間違い
- 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.
Limitations
- This calculator only supports systems of exactly three equations in three unknowns.
活用シーン
- 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.
よくある質問
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.
What is the difference between row echelon form and reduced row echelon form?
In row echelon form, pivots aren't necessarily 1 and values can remain above a pivot. In the reduced form (RREF), every pivot is exactly 1 and every other value in that column is 0, which makes the solution directly readable.
Can I use this for systems with more or fewer than three unknowns?
No, this calculator is specifically built for 3×3 systems. For 2×2 systems you can set two coefficients to 0, but for larger systems you need a different tool.
Why does the result sometimes show a fraction-like number?
Because not every system has whole-number solutions. The calculator shows the result as a decimal, rounded to a few decimal places, even when the exact solution is actually a fraction.
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