Calculator Input
Choose the matrix order, calculation method, and precision. Then fill the matrix cells and calculate the determinant with working steps.
Plotly Graph
This graph shows how the determinant changes when the entire matrix is scaled by a factor k. For an n × n matrix, det(kA) = kn det(A).
Example Data Table
| Example | Matrix | Method Suggestion | Determinant |
|---|---|---|---|
| 2 × 2 Basic | [[4, 7], [2, 6]] | Direct formula | 10 |
| 3 × 3 Worked | [[2, 1, 3], [1, 0, 4], [5, 2, 1]] | Laplace or elimination | 9 |
| 4 × 4 Structured | [[3, 1, 2, 4], [0, 2, 1, 5], [1, 0, 3, 2], [2, 1, 4, 1]] | Elimination | -60 |
| Singular Case | [[1, 2, 3], [2, 4, 6], [1, 0, 1]] | Elimination | 0 |
Formula Used
- 2 × 2 determinant: det(A) = ad - bc.
- Laplace expansion: det(A) = Σ a1j C1j, where C1j = (-1)1+j det(M1j).
- Elimination rule: Row replacement does not change the determinant. Row swaps multiply the determinant by -1.
- Triangular rule: Once the matrix is upper triangular, det(A) equals the product of diagonal entries, adjusted by any row-swap sign.
How to Use This Calculator
- Select a matrix size from 2 × 2 up to 6 × 6.
- Choose Auto, Gaussian elimination, or Laplace expansion.
- Enter each matrix element using integers or decimals.
- Set the desired decimal precision for the displayed steps.
- Click Calculate Determinant to show the answer above the form.
- Review the transformation steps and the scaling graph.
- Use the CSV or PDF buttons to export the current result.
FAQs
1. What does the determinant tell me?
The determinant indicates whether a square matrix is invertible, how area or volume scales under the transformation, and whether rows are linearly independent.
2. Why does a zero determinant matter?
A zero determinant means the matrix is singular. It has no inverse, collapses space in at least one direction, and contains dependent rows or columns.
3. Which method should I choose?
Use Auto for convenience. Laplace expansion is clearer for small matrices, while Gaussian elimination is faster and cleaner for larger matrices.
4. Why do row swaps change the sign?
Each row swap reverses orientation, so the determinant changes sign. Two swaps restore the original sign because their effects multiply together.
5. Can I use decimals and negative values?
Yes. The calculator accepts integers, decimals, and negative entries. Precision settings control how rounded values appear in the final display.
6. Why is the graph based on scaling?
Scaling every matrix entry by k changes the determinant by k raised to the matrix order. The graph makes that relationship easy to inspect.
7. Is Laplace expansion always practical?
Not for large matrices. It becomes lengthy very quickly, so elimination is usually better beyond small classroom examples or symbolic demonstrations.
8. Does this page show full working steps?
Yes. It lists row operations for elimination and detailed cofactor terms for Laplace expansion, making the reasoning behind the result visible.