How this calculation works
Gauss–Jordan elimination swaps rows, scales each pivot to one and clears the other entries in its column. The remaining pivots identify the rank. Nullity counts the free variables in the homogeneous system A x = 0.
rank = pivot count; nullity = number of columns − rankA worked example
For [1 2 3; 2 4 6], the RREF is [1 2 3; 0 0 0]. Rank is 1, nullity is 2 and the only pivot is column 1.
Good to know
For small matrices of exact rational numbers, up to 6 rows and 6 columns. Entries are limited to ±1,000,000; integer fraction denominators to 1,000,000. Decimal input allows up to 6 places. No expressions, scientific notation, complex numbers or uncertain measurements. Large intermediate fractions are bounded to keep the tool responsive.
Reference: University of Nebraska–Lincoln: row echelon forms (opens in a new tab) ↗
Rank counts independent information
A table can contain several rows without containing several independent relationships. In [1 2 3; 2 4 6], the second row repeats the first at twice the scale. Reducing it produces a zero row and leaves rank 1.
Nullity answers a companion question: how many variables can vary freely in A x = 0? The example has three columns and only one pivot, so its nullity is 2. A rectangular matrix can therefore have full row rank while still having nonzero nullity.
Pivot column numbers refer to the original positions
The tool reports pivot column indices starting at 1. If you use the result to choose independent columns of A, take those columns from the original matrix. The reduced matrix helps identify their positions; it is not a substitute for the original column vectors.
Use the displayed operations to reproduce the reduction on paper. A row swap, multiplication by a nonzero factor and subtraction of a multiple of another row preserve the row-equivalent system. Exact fractions make every stage reproducible without a rounding threshold.
Decide whether the last column is data or a constant
This page treats every entered column as part of the matrix. If you paste [A | b], the reported rank and nullity describe that whole augmented matrix. To classify solutions to equations, use the linear system solver, which reserves the final column for constants and reports the coefficient rank separately.
For a homogeneous system, every constant is zero. In that case the all-zero variable vector always satisfies the equations. A positive nullity means additional solutions exist; it does not mean that the homogeneous system is inconsistent.
A few questions, answered
What is the rank of an all-zero matrix?
Zero. It has no pivots, and its nullity equals its number of columns.
Can rank exceed the number of rows?
No. Rank cannot exceed either the row count or the column count.
Will a tiny nonzero value be treated as zero?
No numerical tolerance is applied. An accepted decimal or fraction remains an exact rational value, even if very small. Measurement noise and approximate rank need a different analysis.