Formula Used
Rank counts pivot columns after row reduction. Nullity counts independent free-variable directions. Their sum always equals the column count.
How to Use
Select the required rows and columns. Enter integers, decimals, or fraction values. Press calculate to inspect every result.
- Choose dimensions between one and twelve.
- Select exact or decimal output mode.
- Enter values manually or paste matrix data.
- Enable steps for detailed elimination history.
- Review pivots, bases, properties, and verification.
- Export the finished analysis when needed.
Example Data
These examples show common rank patterns. Nullity always uses the number of columns. Dependent columns increase the free-variable count.
| Matrix type | Rows | Columns | Rank | Nullity |
|---|---|---|---|---|
| Identity matrix | 3 | 3 | 3 | 0 |
| Zero matrix | 3 | 4 | 0 | 4 |
| Dependent columns | 3 | 3 | 2 | 1 |
| Full row rank | 2 | 4 | 2 | 2 |
| Full column rank | 4 | 2 | 2 | 0 |
Key Concepts
Matrix Rank
Rank measures independent rows or columns. It equals the number of pivots. Larger rank means more independent information.
Matrix Nullity
Nullity measures solutions to Ax equals zero. It equals the free-variable count. Each free variable creates one basis direction.
Pivot and Free Variables
Pivot columns contain leading entries in RREF. Other columns represent free variables. Those variables generate the null-space basis.
Common Mistakes
Do not calculate nullity from row count. Avoid rounding too early during elimination. Check fraction signs and zero tolerance carefully.
Applications
Rank detects redundant equations and features. Nullity describes unconstrained solution directions. Both support modelling, graphics, and data analysis.
Frequently Asked Questions
What is matrix rank?
Matrix rank counts independent rows or columns. It equals the pivot count after reduction. Rank cannot exceed either matrix dimension.
What is matrix nullity?
Nullity is the null-space dimension. It equals the number of free variables. Nullity cannot exceed the column count.
Why does nullity use columns?
Each column corresponds to one unknown. Pivot columns determine leading variables. Remaining columns create free variables and nullity.
Can rectangular matrices have nullity?
Yes, rectangular matrices have nullity. The same rank-nullity theorem still applies. Subtract rank from the column count.
What does full column rank mean?
Every column is linearly independent. Rank then equals the column count. The nullity must therefore equal zero.
What does full row rank mean?
Every row is linearly independent. Rank then equals the row count. Extra columns can still create nullity.
Does transpose change matrix rank?
No, transposition preserves matrix rank. Row rank always equals column rank. The calculator verifies this property automatically.
How does tolerance affect results?
Tolerance treats tiny decimals as zero. It helps manage measured or approximate data. Exact mode ignores tolerance for rational zeroes.
When is a square matrix invertible?
A square matrix is invertible at full rank. Its determinant is then nonzero. Its nullity also equals zero.