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RREF Calculator (Reduced Row Echelon Form)

Gauss-Jordan elimination up to 10×10 in exact rational arithmetic, with every row operation written out as R₂ ← R₂ − 2R₁ and the matrix redrawn after each one. Augmented mode classifies the system and gives the solution set.

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RREF Calculator (Reduced Row Echelon Form) is free to use as often as you like, directly from this page. You will find it under Matrices & vectors, with Jordan Normal Form Calculator (J, P and block structure) and Point-Slope Form Calculator for the neighbouring cases.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What is RREF Calculator (Reduced Row Echelon Form)?

Gauss-Jordan elimination up to 10×10 in exact rational arithmetic, with every row operation written out as R₂ ← R₂ − 2R₁ and the matrix redrawn after each one. Augmented mode classifies the system and gives the solution set.

When would I actually use this?

Checking a decomposition done by hand, solving a linear system, and finding the eigenvalues that describe how a transformation stretches space.

What is the most common mistake?

Inverting a matrix to solve a system. Elimination or a factorisation is both faster and far more stable numerically; an explicit inverse amplifies rounding error, especially when the matrix is close to singular.

How is RREF Calculator (Reduced Row Echelon Form) different from Jordan Normal Form Calculator (J, P and block structure)?

They sit next to each other but answer different questions: Jordan Normal Form Calculator (J, P and block structure) is the one to open when you need it to exact Jordan form of a 1×1 to 6×6 matrix: the blocks from the ranks of (A−λI)ᵏ, the transition matrix P built from generalized eigenvector chains, and A·P = P·J checked exactly. When the characteristic polynomial does not split over ℚ it says so instead of guessing. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

Point-Slope Form Calculator is the closest one after this: Write y − y1 = m(x − x1) from a point and slope or two points, with the other line forms.

What else is worth having open alongside it?

Slope-Intercept Form Calculator and Angle Between Two Vectors Calculator — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

The decompositions follow their textbook definitions with partial pivoting where it applies. Results are exact for small integer matrices and subject to floating-point rounding otherwise.

Further reading

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