Calculator Input
Example Data Table
| Matrix | Size | Rank | Rank Deficiency | Nullity | Comment |
|---|---|---|---|---|---|
| [1 2 3; 2 4 6; 1 1 1] | 3 × 3 | 2 | 1 | 1 | Second row depends on the first row. |
| [1 0; 0 1] | 2 × 2 | 2 | 0 | 0 | Identity matrix is full rank. |
| [1 2 3; 2 4 6] | 2 × 3 | 1 | 1 | 2 | Only one independent row remains. |
Formula Used
The calculator uses Gaussian elimination with partial pivoting to transform the matrix into row echelon form. The rank equals the number of pivot rows or pivot columns identified during elimination.
- Rank: number of pivots in row echelon form.
- Maximum possible rank: min(m, n) for an m × n matrix.
- Rank deficiency: min(m, n) − rank(A).
- Nullity: n − rank(A).
- Determinant: shown only for square matrices; a zero determinant often signals rank deficiency.
A tolerance is applied so values close to zero are treated as zero. This helps when the matrix contains decimal values or near-dependent rows.
How to Use This Calculator
- Enter the number of rows and columns for the matrix.
- Set a tolerance level. Smaller values are stricter for pivot detection.
- Type matrix entries row by row using spaces, commas, or semicolons between values.
- Press Submit to display the result above the form.
- Review the rank, deficiency, nullity, pivots, echelon form, and elimination steps.
- Use the export buttons to download the result summary as CSV or PDF.