# Characteristic Polynomial Calculator

URL: https://speedycalculator.com/calculators/math/characteristic-polynomial/
Last updated: 2026-10-30

> The characteristic polynomial calculator computes det(A − λI) (or det(λI − A)) for a 2 × 2, 3 × 3 or 4 × 4 matrix and lists its roots – the eigenvalues.

## Inputs

- Matrix size
- a₁
- b₁
- c₁
- d₁
- a₂
- b₂
- c₂
- d₂
- a₃
- b₃
- c₃
- d₃
- a₄
- b₄
- c₄
- d₄
- Definition

## Results

- Characteristic polynomial
- Coefficients (of det(λI − A), highest power first)
- Roots (eigenvalues)

## Formula

```
p(λ) = det(A − λI)
2 × 2: λ² − tr(A)·λ + det(A)
```

## Explanation

The characteristic polynomial encodes the eigenvalues of a matrix as its roots, and by the Cayley–Hamilton theorem every matrix satisfies its own characteristic polynomial. Its coefficients include the trace and the determinant.

### How to use the Characteristic Polynomial Calculator

- Choose the **matrix size** – the matching grid of fields appears.
- Enter the entries **row by row** (a₁ b₁ c₁ is the first row, a₂ b₂ c₂ the second, and so on).
- Read the result; integer matrices give exact fractions.

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

The coefficients are computed with the Faddeev–LeVerrier algorithm and the roots found numerically. The two common conventions differ only by the sign (−1)ⁿ.

## Worked example

Inputs — Matrix size: 3 × 3; a₁: 4; b₁: 1; c₁: 2; a₂: 1; b₂: 3; c₂: 0; a₃: 2; b₃: 0; c₃: 5; Definition: det(A − λI).

Results — **Characteristic polynomial = det(A − λI) = −λ³ + 12λ² − 42λ + 43**; Coefficients (of det(λI − A), highest power first) = 1, −12, 42, −43; Roots (eigenvalues) = 1.8548973, 3.4760236, 6.6690791.

## Key terms

- **Characteristic polynomial**: det(λI − A), whose roots are the eigenvalues.

## FAQ

**What are the roots of the characteristic polynomial?**
The eigenvalues of the matrix.

**Which convention is correct?**
Both are used; det(λI − A) is monic, det(A − λI) differs by (−1)ⁿ.

**What is the Cayley–Hamilton theorem?**
Substituting A into its characteristic polynomial gives the zero matrix.

**How are the trace and determinant related to it?**
For 2 × 2: λ² − tr(A)λ + det(A).


## Sources

- [Wolfram MathWorld – Characteristic polynomial](https://mathworld.wolfram.com/CharacteristicPolynomial.html)
- [MIT OpenCourseWare – 18.06 Linear Algebra (Gilbert Strang)](https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/)
- [Jim Hefferon – Linear Algebra (free textbook)](https://hefferon.net/linearalgebra/)