Enter Matrix Values
Formula Used
The calculator uses the rank-nullity theorem:
rank(A) + nullity(A) = number of columns of A
Rank is the number of pivot columns after Gaussian elimination. Nullity is the number of free columns. The calculator forms a row echelon matrix, then a reduced row echelon matrix, and reads pivots directly from that structure.
How to Use This Calculator
- Choose the number of rows and columns.
- Click Update Matrix Grid.
- Enter integers, decimals, or simple fractions.
- Set decimal places and tolerance if needed.
- Click Calculate Rank and Nullity.
- Review rank, nullity, pivots, bases, REF, and RREF.
- Use the export buttons to save the result.
Example Data Table
| Matrix | Size | Rank | Nullity | Note |
|---|---|---|---|---|
| [1 2 3; 2 4 6; 0 1 1] | 3 × 3 | 2 | 1 | One dependent column appears. |
| [1 0 2 1; 0 1 1 1; 2 0 4 2] | 3 × 4 | 2 | 2 | Rectangular matrix with two free columns. |
| [1 0 0; 0 1 0; 0 0 1] | 3 × 3 | 3 | 0 | Identity matrix is full rank. |
Matrix Rank and Nullity Guide
Why this topic matters
A matrix rank and nullity calculator helps you study linear algebra faster. Rank measures how many rows or columns are independent. Nullity measures how many free variables remain. These values describe matrix structure, solution freedom, and transformation behavior.
Where rank is used
Rank is useful in equation solving, modelling, coding, and data analysis. Students use it to test system consistency. Engineers use it to inspect model dependence. Analysts use it to detect redundant features before deeper computations.
What this calculator returns
This tool accepts square and rectangular matrices. It supports integers, decimals, and simple fractions. After submission, it applies Gaussian elimination. You get row echelon form, reduced row echelon form, pivot columns, free columns, rank, nullity, and basis vectors.
Why reduced row echelon form helps
Reduced row echelon form makes structure clear. Every pivot becomes one. Every other value in a pivot column becomes zero. That clean shape helps you read independent columns quickly. It also makes null space basis construction much easier.
Understanding the rank-nullity theorem
The rank-nullity theorem connects two important dimensions. Rank counts independent directions created by the matrix. Nullity counts directions sent to zero in the homogeneous system. Their sum always equals the number of columns. This identity is a strong algebra check.
Why nullity is important
Nullity is not only a leftover value. It tells you how many parameters are needed to describe all solutions of Ax = 0. Higher nullity means more freedom in the solution set. Lower nullity means tighter structure and fewer free variables.
Accuracy and tolerance
Some matrices contain decimals. During elimination, tiny rounding values may appear. The tolerance setting tells the calculator when a very small value should be treated as zero. This helps keep pivot detection stable and makes the output easier to interpret.
Better revision workflow
Start with the example matrix. Then enter your own data. Compare the original matrix with REF and RREF. Review pivot columns and basis vectors. Export the result table for notes or assignments. Repeating this process builds stronger intuition for matrix rank and nullity.
FAQs
1) What is matrix rank?
Matrix rank is the number of independent rows or columns. In row reduction, it equals the number of pivot positions. It measures how much unique linear information the matrix contains.
2) What is nullity?
Nullity is the number of free variables in the homogeneous system Ax = 0. It also equals the dimension of the null space. Larger nullity means more solution freedom.
3) How are rank and nullity related?
They are linked by the rank-nullity theorem. Rank plus nullity equals the number of columns in the matrix. This relation is a quick way to verify results.
4) Can I enter fractions?
Yes. The calculator accepts simple fractions such as 1/2 or -3/5. It also accepts integers, decimals, and blank cells, which are treated as zero.
5) Why does the calculator show pivot columns?
Pivot columns identify the independent structure of the matrix. They also show which original columns form a basis for the column space. This is useful for interpretation and study.
6) What does nullity zero mean?
Nullity zero means there are no free variables. The homogeneous system has only the zero solution. In that case, the columns are linearly independent.
7) Why is tolerance included?
Tolerance helps handle tiny rounding errors. During elimination, very small decimal values can appear. The calculator can treat those as zero when they fall below the tolerance level.
8) Can rectangular matrices be used?
Yes. The calculator works with both square and rectangular matrices. Rank and nullity are especially useful for rectangular matrices because free columns often appear clearly there.