Results
Steps
Example data table
Sample 3×3 input and the resulting cofactor matrix. Use “Load example” to insert these values automatically.
| 1 | 2 | 3 |
| 0 | 4 | 5 |
| 7 | 8 | 9 |
| -4 | 35 | -28 |
| 6 | -12 | 6 |
| -2 | -5 | 4 |
Formula used
The cofactor of an element aij is: Cij = (−1)i+j · det(Mij), where Mij is the minor matrix formed by deleting row i and column j.
The cofactor matrix is the matrix of all Cij values for the input. Determinants here are computed by recursive expansion.
How to use this calculator
- Select the matrix size (n×n), then fill the matrix inputs.
- Choose the row i and column j to inspect a specific cofactor.
- Enable “Show steps” to see the minor, determinant, and sign.
- Press “Compute Cofactors” to show results above this form.
- Use Download CSV or Download PDF to export your results.
Professional notes
Cofactors and determinant expansion
A cofactor is the signed minor determinant linked to a specific entry. In Laplace expansion, each matrix element is multiplied by its cofactor, then summed. This relationship makes cofactors useful for validating determinants, building adjugates, and explaining why small numerical changes can shift determinant magnitude. For teaching, reporting cofactors alongside the original matrix helps students see structure rather than memorizing steps.
Minor matrices as local subproblems
Each cofactor requires a minor matrix created by deleting one row and one column. The determinant of that minor captures how the remaining variables interact without the chosen element. In applications, this resembles sensitivity: a large minor determinant indicates strong coupling among the remaining entries. For a 3×3 matrix, every minor is 2×2, so the computational load stays light and results are easy to audit.
Alternating sign pattern and parity
The sign factor (−1)^(i+j) creates a checkerboard of plus and minus entries. This parity rule prevents overcounting when expanding determinants and ensures consistent orientation behavior. Practically, the sign pattern is fixed for a given size, so separating “minor determinant” from “sign” improves error detection. When users see a correct minor but wrong cofactor, the sign is often the culprit.
Performance considerations by size
Recursive determinant evaluation grows rapidly with matrix dimension. A 6×6 cofactor matrix requires many 5×5 determinants, each calling smaller determinants repeatedly. For typical web usage, limiting sizes and encouraging sparse, integer inputs reduces delays. Rounding controls help presentation, but internal calculations should keep full precision until the final display. If performance becomes critical, row reduction or LU methods can replace expansion.
Exportable verification workflow
Engineering and analytics teams often need traceable computation. Exporting the input matrix, full cofactor matrix, and a highlighted selected cofactor supports peer review and documentation. CSV suits spreadsheets and audit logs; PDF suits reports and assignments. When comparing two solutions, confirm the selected minor matrix first, then its determinant, then the applied sign. This staged check isolates mistakes quickly and reliably. In classrooms, exporting both matrices supports grading consistency across sections and reduces disputes. In audits, attaching the PDF alongside source data creates a durable record that can be revisited without rerunning calculations.
FAQs
1) What is a cofactor in a matrix?
A cofactor is a signed value computed from an element’s minor determinant: Cij = (−1)i+j · det(Mij). It is used in determinant expansion and adjugate construction.
2) What is the difference between a minor and a cofactor?
The minor refers to the determinant of the minor matrix det(Mij). The cofactor multiplies that minor by the alternating sign (−1)i+j, producing Cij.
3) Can this compute the entire cofactor matrix?
Yes. It calculates every Cij for the chosen size, then displays the full cofactor matrix and a detailed breakdown for the selected position.
4) Why do results take longer for larger matrices?
Determinants computed by recursive expansion require many sub-determinants. As n grows, the number of operations increases quickly, so 5×5 and 6×6 matrices may feel slower.
5) Should I round results?
Rounding helps readability, especially with decimals. For verification or further calculations, keep rounding off to preserve full precision in the displayed values.
6) How do I verify one cofactor manually?
Delete the chosen row and column to form the minor matrix, compute its determinant, then multiply by (−1)i+j. Compare your result to the selected cofactor output.