Row Reduction (RREF)

Reduce a matrix to row echelon form, step by step

rref(A) · Rank = 3 · Unique solution

Reduced row echelon form

10020103001−1

Rank

Nullity

Pivot columns

1, 2, 3

Free columns

none

Linear system

Unique solution

Steps

1.Scale rowR1 → 1/2·R1
11/2−1/24−3−12−11−212−3
2.EliminateR2 → R2 + 3·R1
11/2−1/2401/21/21−212−3
3.EliminateR3 → R3 + 2·R1
11/2−1/2401/21/210215
4.Scale rowR2 → 2·R2
11/2−1/2401120215
5.EliminateR1 → R1 − 1/2·R2
10−1301120215
6.EliminateR3 → R3 − 2·R2
10−13011200−11
7.Scale rowR3 → -1·R3
10−130112001−1
8.EliminateR1 → R1 + 1·R3
10020112001−1
9.EliminateR2 → R2 − 1·R3
10020103001−1

About this calculator

The Row Reduction Calculator brings any matrix up to 8×8 to reduced row echelon form (RREF) via Gauss-Jordan elimination, listing every swap, scale, and elimination step in standard R-notation. Arithmetic is exact rational (BigInt fractions), so pivots are detected without rounding error and results display as clean fractions like 21/8. It reports the rank and pivot columns, and in augmented mode classifies the linear system as having a unique solution, infinitely many (with the free variables), or none.

How to use the Row Reduction (RREF) calculator

Enter the values for your problem into the input fields.
Read the result — it updates instantly as you type.
Check the formula and the visual explanation to follow how the answer was found.
Copy the page URL to share the exact calculation.

Common examples

System 2x+y−z=8, −3x−y+2z=−11, −2x+y+2z=−3 → RREF in 9 steps: x = 2, y = 3, z = −1
[[2,4],[1,2]] → RREF [[1,2],[0,0]], rank 1, pivot in column 1
Augmented [[1,1,2],[1,1,3]] → row (0 0 | 1) appears: inconsistent, no solution
Augmented [[1,3,1],[2,6,2]] → rank 1, one free variable, infinitely many solutions

Frequently asked questions

How does the Row Reduction (RREF) calculator work?

Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.

When would I use the Row Reduction (RREF) calculator?

It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.

How accurate are the results?

Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.

Can I share a specific calculation with someone else?

Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.