Calculator Form
Example Data Table
This sample 3 × 3 integer matrix returns an exact determinant of 19.
| Row | Column 1 | Column 2 | Column 3 |
|---|---|---|---|
| Row 1 | 2 | 1 | 3 |
| Row 2 | 0 | -1 | 4 |
| Row 3 | 5 | 2 | 0 |
| Determinant | 19 | ||
Formula Used
The determinant of a square matrix A can be defined by Laplace expansion, but this calculator evaluates it with the Bareiss fraction-free elimination method for exact integer output.
The update rule is: aij(k+1) = (akk(k)aij(k) - aik(k)akj(k)) / ak-1,k-1(k-1).
This keeps all intermediate values integral when the original matrix contains integers, which makes the final determinant exact and more stable than ordinary floating-point elimination.
How to Use This Calculator
- Choose a matrix size from 2 × 2 to 6 × 6.
- Enter whole numbers into every matrix cell.
- Use the example button if you want a working sample instantly.
- Click Compute Determinant to display results above the form.
- Review the determinant, principal minors, status, and elimination steps.
- Use the CSV or PDF buttons to save the reported output.
Frequently Asked Questions
1) What matrix sizes are supported here?
This calculator supports square integer matrices from 2 × 2 through 6 × 6. That range keeps the interface readable while still covering most classroom, engineering, and analysis tasks.
2) Why does the form only accept integers?
The page is designed for exact integer determinants. Restricting entries to integers allows fraction-free elimination and avoids rounding drift that can appear when decimals are processed numerically.
3) Which determinant method is used?
The result uses the Bareiss algorithm, an exact elimination method. It behaves like Gaussian elimination but keeps intermediate values integral for integer matrices whenever the required pivots are valid.
4) What does a zero determinant mean?
A zero determinant means the matrix is singular. Its rows or columns are linearly dependent, the matrix is not invertible, and the related linear transformation collapses area or volume.
5) What are leading principal minors?
Leading principal minors are determinants of the top-left 1 × 1, 2 × 2, and larger square blocks. They help reveal structural patterns, sign changes, and stability clues across growing matrix order.
6) Do row swaps affect the determinant?
Yes. Every row swap changes the determinant sign. The calculator tracks the number of swaps and applies the required sign correction to the final value automatically.
7) Can I use this page for invertibility checks?
Yes. A nonzero determinant means the matrix is invertible. A zero determinant means it is singular. This makes determinant testing a quick first check for invertibility.
8) What do the CSV and PDF exports include?
The downloads include the matrix order, determinant, absolute determinant, trace, classification, row-swap count, principal minors, and the entered matrix values shown in the result section.