Calculator Input
Enter each matrix entry as comma-separated coefficients in ascending powers. Example: 3, -2, 1 means 3 - 2x + x2.
Example Data Table
This example is built into the page. Click Load Example to fill the matrix automatically.
| Cell | Coefficient List | Polynomial Form |
|---|---|---|
| A11 | 1,1 | 1 + x |
| A12 | 0,1 | x |
| A13 | 2 | 2 |
| A21 | 2 | 2 |
| A22 | 1,0,1 | 1 + x2 |
| A23 | 1,-1 | 1 - x |
| A31 | 3,0,1 | 3 + x2 |
| A32 | 2,2 | 2 + 2x |
| A33 | 1 | 1 |
Formula Used
Polynomial entry
aij(x) = c0 + c1x + c2x2 + ... + ckxk
Determinant of a polynomial matrix
det(A(x)) = Σσ∈Sn sgn(σ) ∏i=1n ai,σ(i)(x)
Recursive expansion used here
det(A) = Σj=1n (-1)1+ja1j det(M1j)
Point evaluation after the determinant is built
If det(A(x)) = d0 + d1x + ... + dmxm, then det(A(x0)) is found by substituting x = x0.
This page performs exact coefficient-array addition, subtraction, and multiplication first, then forms the full determinant polynomial. That approach is more informative than evaluating entries at one point too early.
How to Use This Calculator
- Select a square matrix size from 2 × 2 up to 5 × 5.
- Enter each matrix entry as a coefficient list in ascending powers.
- Set a specific x value for numeric evaluation.
- Set the plot range and point count for the graph.
- Press Compute Determinant to generate the determinant polynomial.
- Review the summary cards, expanded matrix, coefficient table, evaluations, and Plotly graph.
- Use CSV export for data handling and PDF export for reporting.
Frequently Asked Questions
1) What kind of input does this calculator accept?
It accepts a square matrix whose entries are single-variable polynomials written as comma-separated coefficients in ascending powers. Example: 3, -2, 1 means 3 - 2x + x².
2) Does it compute the determinant polynomial exactly?
Yes. It combines coefficient arrays directly, so the full determinant polynomial is formed before numeric evaluation. Results remain exact within normal floating-point precision for your entered coefficients.
3) Which matrix sizes are practical here?
You can choose 2×2 through 5×5. Larger sizes require more recursive determinant work, so smaller matrices are faster when your entries contain many polynomial terms.
4) Why are coefficients entered in ascending order?
Ascending order maps neatly to array positions: constant first, x next, then x², and so on. That makes polynomial arithmetic transparent and easy to verify.
5) What does det(x) at the chosen value represent?
It is the final determinant polynomial evaluated at your selected x value. Use it to inspect one operating point after the exact symbolic determinant has been formed.
6) Can I use negative numbers and decimals?
Yes. Real coefficients, zero values, and negative numbers are supported. Keep entries comma-separated and avoid typing variable symbols like x or caret notation inside the boxes.
7) What does the Plotly graph show?
The graph plots the determinant polynomial across your chosen x range. It helps reveal sign changes, growth, turning behavior, and flat regions within the selected interval.
8) When should I use CSV and PDF export?
Use CSV when you want coefficient tables and plot points in spreadsheet form. Use PDF when you want a shareable snapshot of the visible result summary and graph.