Row Echelon Form Calculator

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

About Row Echelon Form Calculator

Row reduction uses row swaps, scaling and subtraction to create leading ones with zeros below. The number of pivots is the rank, and the remaining columns correspond to free variables.

Formula

Rank-nullity

rank + number of free columns = total columns

Worked example

With the default values below, Row Echelon Form Calculator returns Row echelon form of [1, 2, 0, 3.3333] [0, 0, 1, 0.3333] [0, 0, 0, 0]. Change any input above and every figure here is recalculated in your browser.

Inputs used

Matrix, row by row: 1, 2, -1, 3, 2, 4, 1, 7, 3, 6, 0, 10
Columns per rowaccepts 2–8: 4

Results

Row echelon form: [1, 2, 0, 3.3333] [0, 0, 1, 0.3333] [0, 0, 0, 0]
Rank: 2
Pivot columns: 1, 3
Free columns (columns minus rank): 2

Step by step

1Values used
2Rank-nullity
3Row echelon form= [1, 2, 0, 3.3333] [0, 0, 1, 0.3333] [0, 0, 0, 0]
4Rank= 2
5Pivot columns= 1, 3
6Free columns (columns minus rank)= 2

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 3 × 4 matrix?

The third row is the sum of the first two, so it reduces away and the rank is 2 with two free columns.

Do row operations change the rank?

No. They preserve the row space, which is exactly why elimination is a valid way to find the rank.