Enter the 3×3 matrix
After submission, the result appears above this form.
Plotly graph
The graph shows p(λ) over a practical viewing range.
Example data table
| Example Matrix | Trace | Determinant | Characteristic Polynomial | Approximate Eigenvalues |
|---|---|---|---|---|
|
[2, 1, 0] [-1, 3, 2] [4, 0, 1] |
6 | 15 | λ³ - 6λ² + 12λ - 15 |
3.912931 1.043534 + 1.656647i 1.043534 - 1.656647i |
Formula used
Characteristic polynomial definition:
p(λ) = det(λI - A)
For a 3×3 matrix A,
p(λ) = λ³ - tr(A)λ² + Sλ - det(A)
Where:
tr(A) = a11 + a22 + a33
S = a11a22 + a11a33 + a22a33 - a12a21 - a13a31 - a23a32
det(A) is the 3×3 determinant.
The roots of p(λ) are the eigenvalues of A.
How to use this calculator
- Enter all nine entries of the 3×3 matrix.
- Click Calculate Polynomial.
- Read the result card above the form.
- Review the trace, minors, determinant, and eigenvalues.
- Inspect the graph to see root behavior visually.
- Download a CSV or PDF summary when needed.
FAQs
1. What does the calculator return?
It returns the 3×3 characteristic polynomial, trace, determinant, sum of principal minors, discriminant, root type, and approximate eigenvalues. It also plots the polynomial on a real-number graph.
2. Which polynomial convention is used?
This page uses p(λ) = det(λI - A). That gives a monic cubic with leading coefficient 1. Some textbooks use det(A - λI), which only changes the overall sign.
3. Are the roots always real?
No. A real 3×3 matrix can have one real eigenvalue and one complex-conjugate pair. The discriminant helps identify whether the cubic has three real roots or mixed root types.
4. Why is the trace important?
The trace is the sum of diagonal entries and also the sum of the eigenvalues. It becomes the coefficient of the λ² term, with a negative sign, in the chosen convention.
5. Why is the determinant shown?
The determinant is the product of the eigenvalues. In this convention, it appears as the constant term with a negative sign. It helps verify the computed polynomial quickly.
6. Can I enter decimals or negative values?
Yes. The inputs accept integers, decimals, and negative numbers. The calculator works with general real-valued 3×3 matrices, not only whole-number examples.
7. What does the graph show?
The graph plots p(λ) against λ on the real axis. Real roots appear where the curve crosses the horizontal axis. Complex roots do not appear as real-axis crossings.
8. When should I use the export buttons?
Use CSV for spreadsheet work or quick tabular records. Use PDF when you need a printable summary for classes, reports, revision files, or documentation.