Calculator Inputs
Choose dimensions, set precision rules, then enter matrix values. Use the example button for a quick test matrix.
Example Data Table
This sample matrix contains one dependent row because the second row is exactly twice the first row. Its rank is 2.
| Row/Col | C1 | C2 | C3 |
|---|---|---|---|
| R1 | 1 | 2 | 3 |
| R2 | 2 | 4 | 6 |
| R3 | 1 | 1 | 1 |
| Expected Rank = 2 | |||
Formula Used
Rank(A) = number of pivot columns in the row-reduced echelon form of matrix A.
Rank(A) = number of non-zero rows in RREF(A).
Nullity(A) = number of columns − rank(A).
For square matrices, a non-zero determinant means the matrix has full rank. This tool performs Gaussian elimination with partial pivoting, normalizes pivot rows, and removes non-pivot entries to obtain RREF.
How to Use This Calculator
- Enter the number of rows and columns for your matrix.
- Set the decimal display precision and a small tolerance value.
- Fill each matrix cell with the required numeric entries.
- Click Calculate Rank to reduce the matrix and view results.
- Review rank, nullity, pivot columns, determinant, trace, and the heatmap.
- Use the export buttons to save a CSV report or PDF summary.
Frequently Asked Questions
1. What does matrix rank mean?
Matrix rank tells you how many independent rows or columns exist in a matrix. It measures the amount of unique linear information present.
2. How does this calculator find rank?
It uses Gaussian elimination to transform the matrix into row-reduced echelon form. The number of pivot positions or non-zero rows gives the rank.
3. What is nullity?
Nullity is the number of free variables in the associated linear system. It equals the total column count minus the rank.
4. Why is tolerance important?
Tolerance decides when a very small number should be treated as zero. This is useful when decimals or rounding create tiny residual values during elimination.
5. Can rank be larger than rows or columns?
No. Rank can never exceed the smaller of the row count or column count. That smaller dimension is the maximum possible rank.
6. What does full rank mean?
A matrix is full rank when its rank equals the smaller matrix dimension. That means it contains the greatest possible number of independent rows or columns.
7. How is determinant related to rank?
For a square matrix, a non-zero determinant means the matrix is invertible and has full rank. A zero determinant indicates rank deficiency.
8. When should I use a matrix rank calculator?
Use it when studying linear systems, vector spaces, machine learning, control models, or any problem where independence, solvability, or dimensionality matters.