Example Data Table
Use this sample 3×3 matrix to see how the adjugate behaves with mixed signs and zeros.
| 2 | -1 | 3 |
| 0 | 4 | 1 |
| 5 | 2 | -2 |
| -10 | 4 | -13 |
| 5 | -19 | -2 |
| -20 | -9 | 8 |
Formula Used
- Minor: Mij is the matrix formed by removing row i and column j.
- Cofactor: Cij = (−1)i+j · det(Mij)
- Adjugate: adj(A) = CT (transpose of the cofactor matrix).
- Inverse (when det ≠ 0): A−1 = adj(A) / det(A)
How to Use This Calculator
- Select the matrix order between 2×2 and 6×6.
- Enter values in the grid; blanks are treated as zero.
- Press Submit to compute cofactors and the adjugate.
- Review the determinant to see if an inverse exists.
- Use Download CSV or Download PDF to save results.
Why adjugate matters in linear algebra
The adjugate links cofactors, determinants, and inverses in one workflow. For any square matrix A, the identity A·adj(A)=det(A)·I holds. This calculator displays det(A) and adj(A) together so you can validate the relationship numerically. When det(A) is zero, the inverse cannot exist, but adj(A) still provides structured information about row and column dependencies that can guide simplification and rank checks.
Supported sizes and practical limits
You can compute adjugates from 2×2 through 6×6, which covers most classroom and engineering examples. Complexity grows rapidly because each cofactor requires a determinant of a smaller matrix. For n=6, the calculator still remains usable for typical numeric inputs, while keeping the process transparent by showing the full cofactor matrix and its transpose. For large n, numerical methods are preferred, but adjugate remains a strong teaching tool.
Determinant as an invertibility test
The determinant is reported with high precision formatting to help distinguish small values from true zeros. If |det(A)| is greater than the tolerance, the tool also returns A⁻¹ using adj(A)/det(A). This is useful for quick verification against other solvers. If det(A)=0, the inverse section is suppressed and the export files clearly label the inverse as unavailable, preventing accidental misuse in downstream calculations.
Interpreting the cofactor and adjugate matrices
Each cofactor Cij combines a minor determinant with the sign pattern (−1)i+j. Large cofactor magnitudes can signal sensitivity: small input changes may cause large changes in det(A) and the inverse. The adjugate is simply Cᵀ, so patterns across rows of C become patterns across columns of adj(A). This view is valuable when checking symbolic work or diagnosing arithmetic slips in hand calculations.
Exports for reporting and review
Results can be downloaded as CSV for spreadsheets and as PDF for printable notes. The exports include the input matrix, cofactor matrix, adjugate matrix, and the determinant. When invertible, the inverse matrix is also included so you can reuse values in modeling, optimization, or verification tasks. This makes the calculator suitable for assignments, lab reports, and documentation where reproducibility matters.
FAQs
1) What is the difference between adjugate and transpose?
Transpose swaps rows and columns. Adjugate is the transpose of the cofactor matrix, so it depends on minors and determinant signs, not only on rearranging entries.
2) Can I compute an inverse using this calculator?
Yes, when det(A) is nonzero. The inverse is computed as A⁻¹ = adj(A)/det(A) and shown below the adjugate section and included in exports.
3) Why is my inverse missing even after submission?
If det(A) equals 0 (or is extremely close to 0), the matrix is singular. The calculator will still show cofactors and adjugate, but it will not display an inverse.
4) How are blanks or invalid inputs handled?
Empty cells or non-numeric values are treated as 0, and the result panel lists input notes so you can correct entries and recompute confidently.
5) Is the adjugate defined for singular matrices?
Yes. Adjugate exists for any square matrix because it is built from cofactors. Singular matrices simply have det(A)=0, which blocks the inverse formula.
6) What file format is best for sharing results?
Use PDF for printing or sending as a report. Use CSV when you want to copy matrices into a spreadsheet, verify formulas, or perform additional numeric analysis.