Calculator Form
Use the responsive controls below. The page stays single-column, while the input controls shift to three, two, or one columns by screen size.
Example data table
| Example | a11 | a12 | a13 | a21 | a22 | a23 | a31 | a32 | a33 | Determinant |
|---|---|---|---|---|---|---|---|---|---|---|
| 3 × 3 sample | 2 | 1 | 3 | 0 | -1 | 4 | 5 | 2 | 1 | 17 |
Formula used
The determinant by minors expands a square matrix along one selected row or column. Each term multiplies an entry by its cofactor.
det(A) = Σ aijCij, where Cij = (-1)i+j det(Mij)Here, Mij is the minor matrix formed by deleting row i and column j. The sign pattern follows a checkerboard rule:
+ - + - ...- + - + ...
+ - + - ...
The calculator computes every needed minor recursively, sums the signed contributions, and reports both the term-by-term expansion and the final determinant.
How to use this calculator
1. Choose the matrix order
Pick a size from 2 × 2 to 6 × 6. The matrix input grid updates automatically.
2. Enter matrix values
Type all values carefully, including negatives or decimals. Empty cells behave like zero when submitted.
3. Select an expansion path
Choose a row or column, or use the recommendation button to target the line with the most zeros.
4. Calculate and review
Submit the form to view the determinant, contribution table, graph, and worked minor matrices above the form.
5. Export results
Use CSV for structured data review or PDF for a neat printable summary with matrix and contribution details.
6. Compare strategies
Try different expansion lines to see how the same determinant appears through different minors and cofactors.
FAQs
1. What does this calculator actually compute?
It computes the determinant of a square matrix by recursive minor expansion. It also shows cofactors, signed term contributions, rank, and a graph for the selected expansion line.
2. Why choose a row or column with zeros?
Zeros remove terms from the expansion. That reduces arithmetic work and makes manual checking easier, especially for larger matrices with many possible minor calculations.
3. Does the determinant change with a different expansion line?
No. Any valid row or column expansion should produce the same determinant. Only the intermediate terms and the amount of work change.
4. What does a zero determinant mean?
A zero determinant means the matrix is singular. Its rows or columns are linearly dependent, and the matrix does not have an inverse.
5. Can I enter decimals and negative values?
Yes. The inputs accept integers, decimals, and negative numbers. Display precision is controlled separately so you can keep the calculations readable.
6. What is shown in the graph?
The graph plots each first-level expansion term as a signed bar. It helps you see which cofactors contribute most to the final determinant.
7. Is this suitable for exam revision?
Yes. The layout is useful for practice because it connects matrix entries, minors, cofactors, signs, and the final determinant in one place.
8. Why is rank included in the result panel?
Rank adds context. Full rank usually matches a non-zero determinant for square matrices, while reduced rank often appears when the determinant is zero.