Calculator Form
Example Data Table
This sample uses column 2 of a 3 × 3 matrix.
| Matrix | Expansion Column | Row | Entry | Minor Determinant | Cofactor | Contribution |
|---|---|---|---|---|---|---|
| [[2, 1, 3], [0, -1, 4], [5, 2, 0]] | 2 | 1 | 1 | -20 | 20 | 20 |
| 2 | -1 | -15 | -15 | 15 | ||
| 3 | 2 | 8 | -8 | -16 | ||
| Final Determinant | 19 | |||||
Formula Used
For a square matrix A, the determinant by column expansion along column j is:
The cofactor for each selected entry is:
Here, Mij is the minor matrix formed by removing row i and column j. The calculator evaluates each term, adds the signed contributions, and returns the determinant.
How to Use This Calculator
- Select a matrix size from 2 × 2 up to 6 × 6.
- Choose the column you want to use for Laplace expansion.
- Enter every matrix value in the grid.
- Set the number of decimal places for displayed results.
- Enable minor matrices if you want detailed learning steps.
- Click Calculate Determinant to show the result above the form.
- Review the term breakdown, contribution graph, and best column suggestion.
- Download the calculation as CSV or PDF when needed.
FAQs
1. What does column expansion mean?
Column expansion is a determinant method that uses one chosen column. Each entry is multiplied by its cofactor, and the signed terms are summed to get the determinant.
2. Why should I choose a column with zeros?
Zeros remove entire terms from the expansion. Fewer active terms make the determinant faster to compute and much easier to check by hand.
3. Can this calculator handle decimal values?
Yes. The input accepts integers, decimals, and negative numbers. The displayed output uses your chosen decimal precision for cleaner reporting.
4. What is the difference between a minor and a cofactor?
A minor is the smaller matrix left after deleting one row and one column. A cofactor adds the correct sign, using the alternating checkerboard pattern.
5. Does the chosen column change the determinant value?
No. Any valid row or column expansion produces the same determinant. Only the number of steps and the convenience of the work change.
6. What matrix sizes are supported here?
This page supports square matrices from 2 × 2 through 6 × 6. That range keeps the step display readable while still covering advanced examples.
7. What does the contribution graph show?
The graph plots each selected-column term. Positive bars increase the determinant, while negative bars reduce it. This helps you see which rows matter most.
8. When is a determinant equal to zero?
A determinant becomes zero when the matrix is singular. This usually happens when rows or columns are dependent, repeated, or linearly related.