Result
Reduced Row Echelon Form
Null Space Basis
Row Reduction Steps
Matrix Input
Example Data Table
| Matrix | Rows | Columns | Rank | Nullity | Meaning |
|---|---|---|---|---|---|
|
[1, 2, -1, 3] [2, 4, 1, 9] [1, 2, 0, 4] |
3 | 4 | 2 | 2 | There are two free variables in the homogeneous system. |
|
[1, 0, 0] [0, 1, 0] [0, 0, 1] |
3 | 3 | 3 | 0 | The columns are independent, so the null space is trivial. |
|
[1, 2, 3] [2, 4, 6] |
2 | 3 | 1 | 2 | Only one pivot column exists, so two variables are free. |
Formula Used
The calculator uses the rank-nullity theorem. For a matrix A with n columns:
Nullity(A) = n - Rank(A)
Rank is found by reducing the matrix to reduced row echelon form. Each pivot column increases rank by one. Columns without pivots become free variable columns. The number of free variables is the nullity.
The null space is the solution set of Ax = 0. If nullity is zero, the matrix has only the trivial zero solution. If nullity is greater than zero, the system has infinitely many homogeneous solutions.
How to Use This Calculator
- Enter the number of rows and columns.
- Click Build Matrix to create input cells.
- Type each matrix value. Fractions and decimals are accepted.
- Set a tolerance if your values include rounding errors.
- Press Find Nullity.
- Review the rank, pivot columns, free variables, and basis vectors.
- Use CSV or PDF buttons to save your report.
Understanding Matrix Nullity
Understanding Matrix Nullity
Matrix nullity measures how many free directions solve Ax = 0. It is the dimension of the null space. A higher value means the matrix loses more information when it transforms a vector. A value of zero means the only solution is the zero vector.
Why Nullity Matters
Nullity connects columns, rank, and linear dependence. In linear algebra, it helps identify whether columns span the input space. It also shows whether a system has unique, infinite, or constrained solutions. Engineers use it for network models. Data analysts use it when features depend on other features. Students use it to understand kernels and transformations.
How the Calculator Works
This calculator first builds the augmented-free matrix from your entries. It then applies row reduction. The tool searches each column for a useful pivot. Rows are swapped when needed. Pivot rows are scaled. Other rows are cleared above and below the pivot. The final reduced form shows rank, pivot columns, and free variable columns.
Reading the Result
Nullity equals the number of columns minus the rank. The number of columns is the input dimension. Rank is the count of pivot columns after row reduction. Free variables are non-pivot columns. Each free variable gives one independent basis vector for the null space.
Best Practices
Enter exact values when possible. Fractions such as 1/2 are accepted. Use decimal values for measured data. Increase tolerance when your matrix contains rounded measurements. Keep tolerance small for exact classroom examples. Check the RREF table before using the answer.
Practical Notes
A square matrix can still have nonzero nullity. That happens when its columns are dependent. A wide matrix often has nullity greater than zero because it has more unknowns than equations. A tall matrix may have zero nullity when all columns are independent.
This page also exports your work. The CSV file is useful for spreadsheets. The PDF report is better for sharing. Use the example table to verify the method. Then replace the values with your own matrix. You can compare several matrices by changing rows, columns, and entries. This makes repeated practice faster, cleaner, and less confusing during exam preparation sessions.
FAQs
What is the nullity of a matrix?
Nullity is the dimension of the null space. It tells how many free variables appear in the solution of Ax = 0.
How is nullity calculated?
Nullity is calculated by subtracting matrix rank from the number of columns. The formula is nullity = columns - rank.
What does nullity zero mean?
Nullity zero means there are no free variables. The homogeneous system has only the zero solution.
Can a square matrix have nonzero nullity?
Yes. A square matrix has nonzero nullity when its columns are linearly dependent and its rank is less than its columns.
What are pivot columns?
Pivot columns are columns that contain leading entries after row reduction. They count toward matrix rank.
What are free variables?
Free variables correspond to non-pivot columns. Each free variable adds one dimension to the null space.
Why does tolerance matter?
Tolerance decides when a very small number should be treated as zero. It helps with rounded decimal data.
Can I download my result?
Yes. Use the CSV button for spreadsheet data or the PDF button for a formatted report.