Characteristic Polynomial Calculator

Enter the four entries of a 2×2 matrix to get its characteristic polynomial p(λ) = det(A − λI), the trace, the determinant, and both eigenvalues.

Quick Facts

Formula
p(λ) = det(A − λI) = λ² − tr(A)λ + det(A)
For a 2×2 matrix, this is always a quadratic in λ.
Eigenvalue link
tr(A) = λ₁ + λ₂ and det(A) = λ₁ · λ₂
The trace and determinant are the sum and product of the eigenvalues.

Your Results

Calculated
Characteristic polynomial
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p(λ) = det(A − λI)
Trace, tr(A)
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a₁₁ + a₂₂ (sum of eigenvalues)
Determinant, det(A)
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a₁₁a₂₂ − a₁₂a₂₁ (product of eigenvalues)
Eigenvalues (λ)
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Roots of the characteristic polynomial

Ready

Enter the matrix entries above and press Calculate.

How the Characteristic Polynomial Calculator works

For any square matrix A, the characteristic polynomial is defined as p(λ) = det(A − λI), where λ (lambda) is a variable, I is the identity matrix of the same size as A, and det denotes the determinant. Setting p(λ) = 0 and solving for λ gives the eigenvalues of A — the scalars for which A behaves like simple scaling along certain directions (eigenvectors).

The formula for a 2×2 matrix

This calculator works with a general 2×2 matrix A = [[a₁₁, a₁₂], [a₂₁, a₂₂]]. Subtracting λ from each diagonal entry and expanding the determinant gives:

  • Characteristic polynomial: p(λ) = λ² − tr(A)λ + det(A)
  • Trace: tr(A) = a₁₁ + a₂₂ (sum of the diagonal entries)
  • Determinant: det(A) = a₁₁a₂₂ − a₁₂a₂₁

Because p(λ) is quadratic in λ, its roots — the eigenvalues — come from the ordinary quadratic formula: λ = [tr(A) ± √(tr(A)² − 4·det(A))] / 2. The quantity under the square root, tr(A)² − 4·det(A), is the discriminant of the characteristic polynomial.

Understanding the output

  • Two distinct real eigenvalues: occurs when the discriminant is positive. This is the most common case for matrices without special symmetry.
  • One repeated eigenvalue: occurs when the discriminant is exactly 0, so λ₁ = λ₂. The matrix may or may not be diagonalizable in this case.
  • Complex-conjugate eigenvalues: occurs when the discriminant is negative. The eigenvalues take the form a ± bi, where i = √(−1). This is typical of rotation matrices.
  • Check your answer: the trace should always equal the sum of the two eigenvalues, and the determinant should always equal their product — a quick way to verify any result by hand.

Common uses

Characteristic polynomials and the eigenvalues they produce are used to analyze the stability of dynamical systems and differential equations, determine whether a matrix is diagonalizable, study long-run behavior of Markov chains, and compute normal modes in physics and engineering. The 2×2 case shown here is also the standard first example taught in linear algebra courses because every step can be verified by hand.

Frequently Asked Questions

What is a characteristic polynomial?
For a square matrix A, the characteristic polynomial is p(λ) = det(A − λI), where λ is a variable and I is the identity matrix. For a 2×2 matrix this simplifies to p(λ) = λ² − tr(A)λ + det(A), using the trace (sum of diagonal entries) and the determinant.
How do you find eigenvalues from the characteristic polynomial?
The eigenvalues of A are the roots of p(λ) = 0. For a 2×2 matrix that is a quadratic equation, so the eigenvalues are λ = [tr(A) ± √(tr(A)² − 4·det(A))] / 2, the same quadratic formula used for any quadratic equation.
What does a negative discriminant mean for the eigenvalues?
The discriminant here is tr(A)² − 4·det(A). If it is negative, the characteristic polynomial has no real roots and the eigenvalues form a complex-conjugate pair a ± bi. This is common for rotation-like matrices and is not an error.
Does this calculator work for matrices larger than 2×2?
No, this tool is built specifically for 2×2 matrices, where the characteristic polynomial is always a quadratic. For an n×n matrix, det(A − λI) expands into a degree-n polynomial, typically found by cofactor expansion or algorithms such as Faddeev-LeVerrier.

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