Characteristic Polynomial Calculator

Background.

Eigenvalues are the numbers that say what a matrix does along its own preferred directions: stretch, shrink, flip or rotate. They decide whether a population model grows or dies out, whether a mechanical system oscillates, and whether an equilibrium of a differential equation is stable. To find them you need the characteristic polynomial first, because the eigenvalues are exactly its roots.

For a 2×2 matrix the polynomial has a pleasingly compact form. Writing out det(λI − A) and expanding gives λ² − (a + d)λ + (ad − bc). The coefficient of λ is minus the trace, the sum of the diagonal entries; the constant term is the determinant. So two numbers you can read off almost at a glance carry everything the polynomial knows, and they have a meaning of their own: the trace is the sum of the two eigenvalues and the determinant is their product.

That link makes the trace and determinant useful even before you solve anything. A negative determinant means the eigenvalues are real and of opposite sign. The quantity tr² − 4·det, the discriminant of the quadratic, tells you whether the eigenvalues are two distinct real numbers, one repeated value, or a complex pair, which is what you get from a matrix that rotates the plane.

This page handles real 2×2 matrices and reports the trace, the determinant and the polynomial built from them. It stops short of solving the quadratic, so the eigenvalues themselves come from applying the quadratic formula to the result. Larger matrices need a determinant expansion or a numerical method and are not covered here.

What is characteristic polynomial calculator?

The characteristic polynomial of a square matrix A is p(λ) = det(λI − A), where I is the identity matrix of the same size and λ is a variable. Its roots are the eigenvalues of A. For a 2×2 matrix it is always a quadratic with leading coefficient 1, whose λ coefficient is minus the trace of A and whose constant term is the determinant of A.

How to use this calculator.

1. 01Write the matrix as [[a11, a12], [a21, a22]], row by row, and enter each entry in the field with the same position label. Swapping a12 and a21 leaves the result unchanged, but moving an entry onto or off the diagonal does not.
2. 02Use decimals for fractions, for example 0.5 rather than 1/2; entries can be negative.
3. 03Read the determinant (the constant coefficient) and the trace, then the polynomial, which is displayed as lambda^2 − (trace)lambda + (determinant) with the signs left inside the brackets.
4. 04To get the eigenvalues, solve the quadratic: λ = (trace ± √(trace² − 4 × determinant)) ÷ 2.

The formula.

For A = [[a, b], [c, d]], det(λI − A) = (λ − a)(λ − d) − bc = λ² − (a + d)λ + (ad − bc). The trace, tr(A) = a + d, is computed first, then the determinant, det(A) = ad − bc, and the polynomial is assembled as λ² − tr(A)λ + det(A). Some textbooks define the polynomial as det(A − λI) instead; for a 2×2 matrix the two definitions give exactly the same quadratic, because swapping the sign of a 2×2 matrix leaves its determinant unchanged. The off-diagonal entries b and c only enter through their product, so they affect the determinant but never the trace. Every coefficient is exact for the entries you type; no eigenvalues are approximated.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.