What is the characteristic polynomial?

For an n×n matrix A, the characteristic polynomial is p(λ) = det(A − λI). Its roots are the eigenvalues of A.

How are eigenvalues related to the polynomial?

Setting p(λ) = 0 gives the characteristic equation. Every solution λ is an eigenvalue — a scalar by which some non-zero vector is scaled when multiplied by A.

Can eigenvalues be complex?

Yes. For real matrices with no real eigenvalues (e.g. a rotation matrix) the roots come as conjugate pairs a ± bi. The calculator displays them in a + bi form.

What does "repeated root" mean?

A repeated root (algebraic multiplicity > 1) means the polynomial has a squared (or higher) factor. The matrix may or may not be diagonalizable depending on the geometric multiplicity.

Why is det(A − λI) used instead of det(λI − A)?

Both give the same roots. The convention det(A − λI) is common in applied math; det(λI − A) is common in pure math. They differ only by a sign factor of (−1)^n, which does not change the roots.

Characteristic Polynomial Calculator

Name: Characteristic Polynomial Calculator
Availability: InStock
Author: Nham Vu

Enter a 2×2 or 3×3 matrix and get its characteristic polynomial, eigenvalue equation, and roots instantly.

Matrix Input

Enter matrix entries (row by row):

A = [ a b ; c d ]

Results

Enter a matrix and click Calculate.

Copied!

Summary

Enter a 2×2 or 3×3 matrix and get its characteristic polynomial, eigenvalue equation, and roots instantly.

How it works

Select whether your matrix is 2×2 or 3×3.
Enter each entry of the matrix into the grid.
Click Calculate to expand det(A − λI) into a polynomial.
The tool displays the polynomial in standard form and solves for the eigenvalue roots.
Use the step-by-step breakdown to follow the cofactor expansion.

Use cases

Find eigenvalues of a matrix in a linear algebra course.
Verify hand-calculated characteristic polynomials.
Explore how matrix entries affect eigenvalue locations.
Check stability of a dynamical system by inspecting roots.
Prepare input for further diagonalization or Jordan-form work.

Frequently Asked Questions

Last updated: 2026-06-10 · Reviewed by Nham Vu