Reduced Row Echelon Form Calculator

Work out reduced row echelon form instantly with clear inputs, formula shown and shareable results.

About Reduced Row Echelon Form Calculator

Reduced row echelon form goes further than plain echelon form: each pivot is 1 and every other entry in its column is 0. For an augmented system the last column then reads off the solution directly.

Formula

Consistency

solvable ⇔ rank of the augmented matrix equals rank of the coefficient matrix

Worked example

With the default values below, Reduced Row Echelon Form Calculator returns Solution of x1 = 2, x2 = 3, x3 = -1. Change any input above and every figure here is recalculated in your browser.

Inputs used

Augmented matrix, row by row: 2, 1, -1, 8, -3, -1, 2, -11, -2, 1, 2, -3
Columns per row (including the constants)accepts 2–8: 4

Results

Solution: x1 = 2, x2 = 3, x3 = -1
Reduced row echelon form: [1, 0, 0, 2] [0, 1, 0, 3] [0, 0, 1, -1]
Rank of the augmented matrix: 3
Pivot columns: 1, 2, 3

Step by step

1Values used
2Consistency
3Solution= x1 = 2, x2 = 3, x3 = -1
4Reduced row echelon form= [1, 0, 0, 2] [0, 1, 0, 3] [0, 0, 1, -1]
5Rank of the augmented matrix= 3
6Pivot columns= 1, 2, 3

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For the sample system?

It reduces to the identity, giving x₁ = 2, x₂ = 3, x₃ = −1.

How do I spot no solution?

A row of zeros with a non-zero constant means 0 = c, which is impossible, and shows up as the augmented rank exceeding the coefficient rank.