Matrix Rank Calculator

Determine the rank of a matrix by row-reducing to echelon form.

About Matrix Rank Calculator

The rank of a matrix is the number of linearly independent rows (or equivalently columns). It equals the number of pivot positions in the reduced row echelon form. The rank-nullity theorem states: rank + nullity = number of columns.

Formula

Rank-Nullity Theorem

rank(A) + nullity(A) = n (number of columns)

A: Matrix
n: Number of columns

Worked example

With the default values below, Matrix Rank Calculator returns Rank of 2. Change any input above and every figure here is recalculated in your browser.

Inputs used

Matrix A (rows separated by newlines or semicolons; values by commas or spaces): 1 2 3 4 5 6 7 8 9

Results

Rank: 2
Reduced Row Echelon Form: 1 0 -1 0 1 2 0 0 0
Pivot Columns: 1, 2
Nullity: 1

Step by step

1Original Matrix= 1 2 3 4 5 6 7 8 9
2RREF= 1 0 -1 0 1 2 0 0 0
3Pivot columns (1-indexed)= 1, 2
4Rank= 2
5Nullity (cols − rank)= 1

Assumptions and limits

The result is derived only from the single input 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

What does rank tell us about a system of equations?

If rank equals the number of unknowns, the system has a unique solution. If rank is less, the system has infinitely many solutions or is inconsistent.

Can the rank exceed the number of rows or columns?

No. The rank is at most min(m, n) for an m×n matrix.