BYTETOOLS

RREF Calculator

Reduce any matrix to reduced row echelon form with every row operation listed, pivot columns marked, and the rank and consistency of the system reported.

3×4
Size
3
Rank
1, 2, 3
Pivot columns
11
Row operations

Reduced row echelon form

100-8
0101
001-2

Pivot columns are highlighted. Every pivot is 1 and is the only non-zero entry in its column — that is what makes this form reduced.

System verdict

Consistent with exactly one solution — every variable column holds a pivot, and the last column is that solution.

Row operations performed (11)

  1. 1.R1 ↔ R2
  2. 2.R1 ← (1/2)·R1
  3. 3.R2 ← R2 − (1)·R1
  4. 4.R3 ← R3 − (-2)·R1
  5. 5.R2 ↔ R3
  6. 6.R2 ← (1/3)·R2
  7. 7.R1 ← R1 − (3/2)·R2
  8. 8.R3 ← R3 − (1/2)·R2
  9. 9.R3 ← (1/1/6)·R3
  10. 10.R1 ← R1 − (3/2)·R3
  11. 11.R2 ← R2 − (-4/3)·R3

Method: Gauss-Jordan elimination with partial pivoting, carried out in exact rational arithmetic — results such as 2/3 stay as fractions instead of rounding to 0.6666667.

What is the RREF Calculator?

The ByteTools RREF Calculator puts any matrix up to 10×10 into reduced row echelon form using Gauss-Jordan elimination with partial pivoting.

  • Exact fraction arithmetic — answers print as 2/3, never as 0.6666667
  • Every row operation logged in Rᵢ ↔ Rⱼ, Rᵢ ← kRᵢ, Rᵢ ← Rᵢ − mRⱼ notation
  • Pivot columns highlighted, with rank and nullity reported
  • Augmented mode detects unique, infinite and inconsistent systems
  • Accepts integers, decimals like −2.5 and fractions like 3/4
  • All computation is local to your browser

How to use the RREF Calculator

  1. 1

    Type your matrix one row per line, with values separated by spaces or commas.

  2. 2

    Tick or untick 'Last column is the right-hand side' depending on whether you entered an augmented system.

  3. 3

    Read the reduced row echelon form, where highlighted columns are the pivots.

  4. 4

    Check the system verdict for a unique, infinite or non-existent solution.

  5. 5

    Expand the row-operation list to see exactly how the reduction was carried out, and copy it.

About the RREF Calculator

The ByteTools RREF Calculator puts any matrix up to 10×10 into reduced row echelon form using Gauss-Jordan elimination with partial pivoting. Crucially, it does the arithmetic in exact fractions rather than decimals, so a pivot of 3 gives you 2/3 instead of 0.6666667 — the answers match what you would get working the problem by hand.

Every row operation is logged in standard notation: Rᵢ ↔ Rⱼ for a swap, Rᵢ ← kRᵢ for a scale, and Rᵢ ← Rᵢ − mRⱼ for an elimination. Pivot columns are highlighted in the result, and the rank is the number of pivots. Tick the augmented-matrix option and the tool also reads off whether the system has exactly one solution, infinitely many, or none at all — the last case being a row that reduces to 0 = a non-zero number.

This is a study tool for linear algebra students checking homework and for anyone solving systems of equations by hand. It runs entirely in your browser using JavaScript, so nothing is uploaded, nothing is stored, and it keeps working with no connection.

Frequently asked questions

What is reduced row echelon form?

A matrix is in reduced row echelon form when each leading entry is a 1, each leading 1 is the only non-zero value in its column, leading 1s move to the right as you go down, and any all-zero rows sit at the bottom. Every matrix has exactly one RREF, which is why it is such a useful canonical form.

What is the difference between row echelon form and reduced row echelon form?

Row echelon form only requires zeros below each pivot, and pivots need not be 1. Reduced row echelon form goes further: pivots are 1 and every other entry in a pivot column is cleared to zero as well. REF is not unique, RREF is.

How does RREF tell you if a system has no solution?

Look for a row where all the coefficients are zero but the constant on the right is not — that row literally says 0 = 5, which is impossible. When you tick the augmented option, this calculator finds that row and names it for you.

What are pivot columns and free variables?

A pivot column contains a leading 1 in the reduced form; its variable is determined by the others. Any column without a pivot is a free variable that can take any value. The number of free variables is the nullity, and it is what makes a consistent system have infinitely many solutions.

Why does this calculator use fractions instead of decimals?

Because Gauss-Jordan elimination on whole numbers produces exact rational answers, and rounding them to decimals both loses precision and stops the result matching your hand-worked version. Exact fractions also mean a value is either genuinely zero or genuinely not, with no floating-point ambiguity.

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