Matrix inputs
Fill the 4×4 grid. Values can be integers or decimals.
Example data table
This sample matrix has a known determinant of 24.
| Row | c1 | c2 | c3 | c4 |
|---|---|---|---|---|
| r1 | 1 | 2 | 0 | 1 |
| r2 | 3 | 4 | 1 | 2 |
| r3 | 2 | 1 | 3 | 0 |
| r4 | 0 | 2 | 5 | 1 |
Formula used
The determinant of a 4×4 matrix A can be defined by cofactor expansion (Laplace expansion):
det(A) = Σ (-1)^{i+j} · a_{ij} · det(M_{ij}), where M_{ij} is the 3×3 minor formed by removing row i and column j.
This tool computes the same value using Gaussian elimination with partial pivoting, which is efficient and numerically stable for many inputs.
How to use this calculator
- Enter sixteen values into the 4×4 input grid.
- Pick a rounding option that matches your reporting needs.
- Enable steps if you want to review row operations.
- Press Calculate determinant to compute det(A).
- Use the download buttons to export CSV or PDF.
FAQs
1) What does a zero determinant mean?
A zero determinant means the matrix is singular. It has no inverse, and its rows or columns are linearly dependent.
2) Why can rounding change the result?
Nearly singular matrices amplify tiny numerical errors. Rounding hides small pivot differences, so the final product can shift. Increase decimals for sensitivity checks.
3) Which method is used here?
The calculator uses Gaussian elimination with partial pivoting. It tracks row swaps and multiplies the resulting pivot values to obtain the determinant.
4) Can I use negative or decimal values?
Yes. Any real numeric input works, including negatives and decimals. Use a higher rounding setting to preserve detail.
5) How do row swaps affect the determinant?
Each swap flips the determinant’s sign. The tool records swaps and applies a ± sign multiplier to the pivot product.
6) What if my matrix is ill-conditioned?
Ill-conditioned matrices can produce unstable results under finite precision. Try scaling values, increasing decimals, and comparing outcomes to spot sensitivity.
7) Are the CSV and PDF exports identical?
Both contain the determinant and the matrix values. CSV is best for spreadsheets, while PDF is handy for sharing a fixed report.