Build determinants from matrices with guided calculations. Review minors, cofactors, and row expansion details instantly. Export answers, steps, and examples for study or sharing.
| Size | Sample Matrix | Determinant |
|---|---|---|
| 2 × 2 | [[4, 7], [2, 6]] | 10 |
| 3 × 3 | [[2, 1, 3], [0, -1, 4], [5, 2, 0]] | 19 |
| 4 × 4 | [[1, 2, 0, 1], [0, 3, 1, 2], [0, 0, 4, 1], [0, 0, 0, 5]] | 60 |
The calculator finds the determinant by converting the matrix into an upper triangular form. Row replacement keeps the determinant unchanged. A row swap changes the sign. After elimination, the determinant equals the signed product of the diagonal entries.
Core rule: det(A) = (-1)s × d1 × d2 × ... × dn, where s is the number of row swaps and each d is a diagonal entry of the upper triangular matrix.
This result matches cofactor expansion and minor-based methods, but elimination is faster and cleaner for larger matrices. The graph in this tool follows the cumulative pivot product so you can see how the determinant develops across the main elimination stages.
Choose a square matrix size from 2 × 2 up to 5 × 5. Enter every matrix value in the input grid. You can also click Load Example to place a ready-to-test matrix into the form.
Set the decimal precision to control how the numbers appear in the steps and final result. Leave partial pivoting enabled for more stable elimination when your matrix contains zero or very small pivot values.
Press Calculate Determinant. The page will show the determinant immediately below the header and above the form, followed by the transformed matrix, a progress graph, and every elimination step. Use the CSV and PDF buttons to export the result.
This tool works with square matrices from 2 × 2 to 5 × 5. It accepts integers, decimals, negatives, and zeros.
Swapping two rows reverses matrix orientation. Because determinant measures signed scaling, each row exchange multiplies the determinant by negative one.
No. Replacing one row with itself minus a multiple of another row keeps the determinant unchanged. That is why elimination is efficient here.
Yes. The inputs accept any numeric values supported by the browser, including fractions entered as decimals and signed numbers.
A zero determinant means the matrix is singular. During elimination, this appears when a pivot column cannot produce a nonzero pivot.
Partial pivoting moves the largest available entry into the pivot position. This reduces unstable division and handles zero pivots more safely.
For upper triangular matrices, the determinant equals the product of diagonal entries. Elimination converts the matrix into that form while tracking sign changes.
Yes. You can check small matrices with cofactor expansion, minors, or Sarrus for 3 × 3 cases. The result should match this calculator.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.