Characteristic polynomial guides
Read these articles in order for a complete learning path. Each guide is written for students who need to calculate, simplify, and apply characteristic polynomials in linear algebra, engineering mathematics, and exam preparation. Start with definitions, follow the formula and step-by-step expansion guides, learn the calculator workflow, study worked examples and eigenvalues, then finish with characteristic equations, determinants, minimal polynomials, and Cayley-Hamilton. Every article includes contextual links to related guides and the home page calculator.
All articles
- Cayley-Hamilton Theorem Explained with Examples
Cayley-Hamilton theorem: substitute A into p(λ) so p(A) = 0. Worked 2×2 matrix example, matrix powers, and links to minimal polynomial and exam-style checks.
- Characteristic Polynomial vs Minimal Polynomial
Compare characteristic and minimal polynomials: definitions, degrees, shared eigenvalues, when m(λ) divides p(λ), and what each polynomial tells you in advanced linear algebra.
- Determinants and Characteristic Polynomials Explained
See how determinant rules shape p(λ): polynomial degree, constant term det(A), singular matrices, trace relations, and invariants you can read before full expansion.
- Characteristic Equation Calculator: Solve p(λ) = 0
Set up and solve the characteristic equation from p(λ) = det(A − λI). Factor quadratics and cubics, list eigenvalues with multiplicity, and know when numerical solvers apply.
- Characteristic Polynomial and Eigenvalues Explained
How characteristic polynomials connect to eigenvalues: roots of p(λ), algebraic multiplicity, diagonalization, and the spectral meaning of det(A − λI) = 0.
- Characteristic Polynomial Examples for 2×2 and 3×3
Worked characteristic polynomial examples for 2×2 and 3×3 matrices. See expansion steps, factoring p(λ), eigenvalue roots, and how to verify each result with the calculator.
- How to Use the Characteristic Polynomial Calculator
Use the free characteristic polynomial calculator for 2×2, 3×3, and 4×4 matrices. Learn what to enter, det(A − λI) vs det(λI − A), and how to verify expansions by hand.
- How to Find the Characteristic Polynomial (Step by Step)
Learn how to find p(λ) step by step: build A − λI, expand det(A − λI), simplify the polynomial, and connect roots to eigenvalues. Includes checks and calculator tips.
- Characteristic Polynomial Formula: det(A − λI) Explained
Characteristic polynomial formulas for p(λ) = det(A − λI): identity matrix setup, expansion rules, 2×2 shortcuts, and how trace and determinant appear in coefficients.
- What Is a Characteristic Polynomial? Definition and Meaning
What is a characteristic polynomial? Learn p(λ) = det(A − λI), the matrix polynomial meaning, eigenvalue links, and how p(λ) fits linear algebra and engineering coursework.