AlgebraFree

Characteristic Polynomial Calculator

Use this free characteristic polynomial calculator to find p(lambda) for any 2x2 matrix. Get the polynomial coefficients, trace, determinant, and roots (eigenvalues) with a step-by-step breakdown.

Enter Values

a (Row 1, Col 1)

Top-left entry of the matrix

b (Row 1, Col 2)

Top-right entry of the matrix

c (Row 2, Col 1)

Bottom-left entry of the matrix

d (Row 2, Col 2)

Bottom-right entry of the matrix

Result

Enter values above and click Calculate to see your result.

AI Assistant

Ask about this calculator

I can help you understand the characteristic polynomial calculator formula, interpret your results, and answer follow-up questions.

Try asking

Formula

p(lambda) = lambda squared - trace(A) lambda + det(A)

For a 2x2 matrix, the characteristic polynomial is always a quadratic. The coefficient of lambda is the negative trace (sum of diagonal entries), and the constant term is the determinant. The roots of this polynomial are the eigenvalues.

Worked Example

A = [[4, 1], [2, 3]] Step 1: Trace = 4 + 3 = 7 Step 2: Determinant = (4)(3) - (1)(2) = 10 Step 3: Characteristic polynomial: p(lambda) = lambda squared - 7 lambda + 10 Step 4: Factor: (lambda - 5)(lambda - 2) = 0 Step 5: Roots: lambda = 5 and lambda = 2 Result: p(lambda) = lambda squared - 7 lambda + 10, with eigenvalues 5 and 2.

What Is the Characteristic Polynomial?

The characteristic polynomial of a matrix A is defined as det(A - lambda I), where I is the identity matrix and lambda is a variable. For a 2x2 matrix, this always produces a quadratic polynomial. The roots of the characteristic polynomial are the eigenvalues of the matrix, making it the bridge between matrix algebra and polynomial algebra.

For a 2x2 matrix, the characteristic polynomial is always quadratic: lambda squared - (trace) lambda + (determinant)
The roots of the characteristic polynomial are exactly the eigenvalues of the matrix
The trace equals the sum of eigenvalues, and the determinant equals their product (Vieta's formulas)
The discriminant (trace squared - 4 det) determines whether eigenvalues are real, repeated, or complex
The Cayley-Hamilton theorem states that every matrix satisfies its own characteristic polynomial: p(A) = 0

The characteristic polynomial is the standard method for finding eigenvalues analytically. For 2x2 and 3x3 matrices, the polynomial can be solved exactly. For larger matrices, numerical methods are used instead.

You can also calculate changes using our Eigenvalue Calculator, Eigenvector Calculator, 2x2 Determinant Calculator or Diagonalization Calculator.

Frequently Asked Questions

What are the roots of the characteristic polynomial?

The roots are the eigenvalues of the matrix. For a 2x2 matrix, the characteristic polynomial is quadratic, so there are exactly two roots (counting multiplicity). They can be found using the quadratic formula.

Why does the trace appear as a coefficient?

The trace (sum of diagonal entries) is always equal to the sum of eigenvalues. By Vieta's formulas for quadratics, the sum of roots equals the negative of the linear coefficient, which is the trace.

What does the discriminant tell me?

The discriminant (trace squared - 4 det) determines the nature of eigenvalues. Positive means two distinct real eigenvalues, zero means a repeated real eigenvalue, and negative means two complex conjugate eigenvalues.

What is the Cayley-Hamilton theorem?

The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic polynomial. If p(lambda) = lambda squared - 7 lambda + 10, then A squared - 7A + 10I = 0. This can be used to express A inverse and higher powers of A in terms of A and I.

Can I find the characteristic polynomial for larger matrices?

Yes, but the polynomial degree equals the matrix size. A 3x3 matrix has a cubic characteristic polynomial, a 4x4 has quartic, and so on. For large matrices, computing the polynomial analytically becomes impractical and numerical eigenvalue methods are preferred.

How does the characteristic polynomial relate to the determinant?

The constant term of the characteristic polynomial (when lambda = 0) equals the determinant of A. This is because det(A - 0 I) = det(A). The determinant is also the product of all eigenvalues.

AI Assistant

Ask about this calculator

I can help you understand the characteristic polynomial calculator formula, interpret your results, and answer follow-up questions.

Try asking

Related Calculators

Eigenvalue Calculator Eigenvector Calculator 2x2 Determinant Calculator Diagonalization Calculator

More Algebra Calculators

View all

Algebra Graphing Calculator

Plot 2D functions and algebraic equations.

Inequality Graphing Calculator

Graph and shade mathematical inequalities.

Function Transformation Calculator

Visualize and calculate function transformations.

Completing the Square Calculator

Convert quadratics to vertex form step by step.

Explore Other Categories

Accurate and Reliable

All calculations run locally. Solve equations with confidence using AI-verified methods.