Characteristic Polynomial Calculator

Find the characteristic polynomial det(A - λI) = 0 for 2x2 and 3x3 matrices with step-by-step determinant expansion.

Enter Matrix Elements

Matrix A (2x2):

Result

Characteristic Polynomial
--
p(λ) = 0
Trace (sum of diagonal) --
Determinant --
Eigenvalues --

Step-by-Step Solution

det(A - λI) = 0

What Is the Characteristic Polynomial?

The characteristic polynomial of a square matrix A is a polynomial p(lambda) defined by the equation det(A - lambda * I) = 0, where I is the identity matrix and lambda is a scalar variable. The roots of the characteristic polynomial are the eigenvalues of the matrix. The characteristic polynomial encodes key information about the matrix, including its trace, determinant, and eigenvalues.

For an n x n matrix, the characteristic polynomial has degree n. It is one of the most fundamental objects in linear algebra and has applications across mathematics, physics, engineering, and computer science.

Characteristic Polynomial Formulas

2x2 Matrix

For A = [[a,b],[c,d]], the characteristic polynomial is a quadratic.

p(L) = L^2 - (a+d)L + (ad-bc)

3x3 Matrix

For a 3x3 matrix, the polynomial is cubic with trace, cofactor sum, and determinant.

p(L) = -L^3 + tr(A)L^2 - ... + det(A)

Eigenvalues

The eigenvalues are the roots of p(lambda) = 0.

det(A - lambda*I) = 0

Cayley-Hamilton Theorem

Every square matrix satisfies its own characteristic polynomial.

p(A) = 0

How to Find the Characteristic Polynomial

  1. Form (A - lambda*I): Subtract lambda from each diagonal entry of the matrix A.
  2. Compute the determinant: Expand det(A - lambda*I) using cofactor expansion or the rule of Sarrus (for 3x3).
  3. Simplify: Collect terms to write the polynomial in standard form.
  4. Find eigenvalues: Solve p(lambda) = 0 for lambda. For 2x2, use the quadratic formula. For 3x3, factor or use numerical methods.

Trace and Determinant

Two important properties of the characteristic polynomial relate to the matrix trace and determinant. The trace of A (the sum of diagonal elements) equals the sum of the eigenvalues, while the determinant of A equals the product of the eigenvalues. For a 2x2 matrix [[a,b],[c,d]], the characteristic polynomial is lambda^2 - (a+d)*lambda + (ad - bc).

Applications

  • Stability Analysis: In control theory, the eigenvalues determine system stability. If all eigenvalues have negative real parts, the system is stable.
  • Principal Component Analysis: PCA uses eigenvalues of the covariance matrix to identify the most significant dimensions in data.
  • Quantum Mechanics: The eigenvalues of Hermitian matrices correspond to observable measurement outcomes.
  • Graph Theory: The characteristic polynomial of the adjacency matrix encodes structural properties of the graph.
  • Differential Equations: Systems of linear differential equations are solved by finding the eigenvalues of the coefficient matrix.

Special Cases

  • Diagonal matrices: The characteristic polynomial factors as a product of (lambda - d_i), where d_i are the diagonal entries.
  • Triangular matrices: The eigenvalues are simply the diagonal entries.
  • Symmetric matrices: All eigenvalues are real numbers.
  • Orthogonal matrices: All eigenvalues have absolute value 1.