Nullity Calculator
Nullity is the number of free variables hiding inside a matrix: columns minus rank. Type any matrix from 2-by-2 up to 6-by-6 into the grid below and this calculator runs Gaussian elimination, reports the rank and the nullity, and checks the rank-nullity theorem on the spot.
Rank is found by Gaussian elimination with partial pivoting; values smaller than 0.0000000001 count as zero during the elimination. Nullity is then columns minus rank.
Quick Answer
For the pre-filled 3-by-3 matrix [[1, 2, 3], [4, 5, 6], [7, 8, 9]], elimination finds 2 pivots, so the rank is 2 and the nullity is 3 – 2 = 1. That one free variable means the homogeneous system Ax = 0 has a one-dimensional solution space. Change any entry and click Calculate nullity to run your own matrix.
Key takeaways
- Nullity = number of columns – rank. It is never negative, and it is always a whole number.
- Rank is the number of pivots Gaussian elimination finds; every other column is a free variable.
- Nullity 0 means the columns are linearly independent, so Ax = 0 has only the trivial solution x = 0.
- Blank cells count as zero, decimals are fine, and values smaller than 0.0000000001 are treated as zero during elimination.
- Changing the row or column count rebuilds the entry grid, so set the size before typing your entries.
How to use the nullity calculator
Set the size, fill the grid, click Calculate nullity, and read three things: the rank, the nullity, and the verdict badge. The results stay hidden until you calculate.
- Set the rows and columns: any size from 2-by-2 up to 6-by-6. The grid rebuilds when the size changes, which clears whatever you typed, so pick the size first. A 3-by-3 with the worked example is pre-filled.
- Type the entries: whole numbers or decimals, one per cell. A blank cell counts as zero, so you can leave the cells you do not need empty.
- Click Calculate nullity: the tool runs Gaussian elimination with partial pivoting, counts the pivots for the rank, and subtracts from the column count for the nullity.
- Read the verdict badge: green means full rank (only the trivial solution), amber names the dimension of the null space and how many free variables that gives you. The theorem box underneath shows the arithmetic check, rank + nullity = columns.
Reset restores the 3-by-3 worked example and hides the results. Copy my results puts the rank, nullity, and verdict on your clipboard; Share results opens your device’s share sheet, or copies the text where sharing is not available.
The rank-nullity formula, in one line
Nullity is not measured directly. The calculator finds the rank first, then applies the rank-nullity theorem: nullity = n – rank, where n is the number of columns.
rank(A) = number of pivots in the row-echelon form of A = dimension of the column space
nullity(A) = number of free variables = dimension of the null space {x: Ax = 0}
The theorem holds for every matrix over the real numbers, and it is proved by counting pivots: each pivot column contributes one dimension to the column space, and each non-pivot column contributes one free variable to the null space, so the two counts always add up to the number of columns. The Dartmouth linear algebra companion states and proves the theorem in full, and the Carleton MATH 1107 notes give the pivot-counting proof from reduced row-echelon form. The elimination method itself, the row operations the calculator performs internally, is set out step by step in the OpenStax guide to Gaussian elimination.
One practical note: because the arithmetic is floating point, the calculator treats any value smaller than 0.0000000001 as zero while eliminating. That keeps tiny rounding remnants from being mistaken for pivots, which is the standard way numerical tools avoid phantom rank.
Worked example: a 3-by-3 matrix with one redundant row
The pre-filled matrix is [[1, 2, 3], [4, 5, 6], [7, 8, 9]]. Notice the third row: it equals 2 times the second row minus the first (2[4,5,6] – [1,2,3] = [7,8,9]), so one row is redundant and the rank cannot be 3. Elimination confirms it:
Two pivots mean two independent columns and one free variable. The theorem box on the calculator shows the check: Rank + Nullity = 2 + 1 = 3, matching the column count, with the green-or-amber verdict badge reading “Null space of dimension 1: 1 free variable in the solution of Ax = 0.” Try the identity matrix next: three pivots, rank 3, nullity 0, and the badge flips to full rank.
Reading your result: what rank and nullity say about a matrix
The two numbers describe the shape of the matrix’s information. Every row below is a real run of the calculator above, so type any of these matrices in and you will get the same figures back.
| Matrix | Rank | Nullity | What it means |
|---|---|---|---|
| 3-by-3 identity | 3 | 0 | Full rank Every column pulls its weight; Ax = 0 has only x = 0. |
| 3-by-3 zero matrix | 0 | 3 | Maximal nullity Nothing survives; every vector maps to zero. |
| [[1,2,3],[4,5,6],[7,8,9]] | 2 | 1 | One free variable One dependent column; a one-dimensional solution space. |
| 2-by-4, second row double the first | 1 | 3 | Wide matrix A wide matrix always has nullity of at least columns – rows. |
| 4-by-2 with independent columns | 2 | 0 | Full rank Tall and independent; still only the trivial solution. |
Two patterns are worth memorizing. A square matrix has nullity 0 exactly when it is invertible, which is the same condition the inverse matrix calculator tests from the other direction. And for a 2-by-2, rank loss shows up as a zero determinant, which is why the matrix determinant calculator and this page tell the same story in different numbers. When you move from one matrix to solving a system, the nullity becomes the count of free variables in the solution, the number the linear systems calculator leaves undetermined.
What competing calculators miss
Rank and nullity tools fall into a few camps, and each leaves something out:
- Step-by-step rank tools (OnlyCalculators): show every row operation and update as you type, which is lovely for learning, but they cap matrices at 4-by-4 and the live-updating layout can feel jumpy on a phone.
- Formula pages (Calculator Ultra): the math is correct and the example is the same classic 3-by-3, but the page is formula-first with no interactive entry grid to experiment in.
- Null space specialists (Omni Calculator): the best of the bunch for learning, because they show the actual basis vectors of the null space, not just its dimension. Still capped at 4-by-4.
- Theory-heavy pages (Brave Calculator, CalculatorsHub): solid definitions and properties, thinner on the tool itself.
- Guides without tools (CopyProgramming-style roundups): useful reading, but you still need a calculator open in another tab.
- The gap this page fills: matrices up to 6-by-6 instead of the usual 4-by-4 ceiling, an explicit rank-nullity theorem check printed with every result, a plain-English verdict badge naming the free-variable count, blank-safe entries, and copy and share buttons so the result can leave the page. Rank itself is the count of linearly independent columns, and seeing the nullity beside it makes that idea concrete instead of abstract.
Frequently asked questions
Nullity is the dimension of the matrix’s null space: the number of independent vectors x that satisfy Ax = 0. In practical terms, it is the number of free variables in the system, the columns that carry no new information beyond what the other columns already say.
Nullity = (number of columns) – rank. Find the rank by Gaussian elimination, counting the pivots, then subtract from the column count. The rank-nullity theorem guarantees the two always add up to the number of columns.
Rank counts the independent columns, the dimensions where the matrix actually does something. Nullity counts the rest: the dependent columns and the free variables. A 3-column matrix with rank 2 has nullity 1, so one column is redundant and the solution space has one free direction.
The columns are linearly independent, so Ax = 0 has only the trivial solution x = 0. For a square matrix, nullity zero is exactly the condition for being invertible. The calculator shows a green full-rank badge in this case.
No. Rank can never exceed the number of columns, so columns minus rank is always zero or positive. If a tool ever reports a negative nullity, its rank computation is wrong.
Any size from 2-by-2 up to 6-by-6. That ceiling is higher than the 4-by-4 cap on several competing tools, and it keeps the elimination fast enough to run instantly in the browser.
Changing the row or column count rebuilds the whole entry grid, which clears what you typed. Set the size first, then enter your values. This matches how the grid is constructed: one table per size.
A variable that is not tied to a pivot column after elimination, so it can take any value while the pivot variables adjust to match. Each free variable adds one dimension to the null space, which is why the free-variable count and the nullity are the same number.
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References
- Dartmouth College, Department of Mathematics. The rank-nullity theorem, Undergraduate Linear Algebra Companion. Accessed October 23, 2026.
- Carleton University, School of Mathematics and Statistics. Rank-Nullity Theorem, MATH 1107 course notes. Accessed October 23, 2026.
- Massachusetts Institute of Technology, Open Learning Library. 18.06SC Linear Algebra (Gilbert Strang). Accessed October 23, 2026.
- nLab. Rank-nullity theorem. Accessed October 23, 2026.
- OpenStax. Solving Systems with Gaussian Elimination, College Algebra with Corequisite Support 2e. Accessed October 23, 2026.