Quick answer
Expand det(A − λI), simplify, factor when possible, and read roots as eigenvalues.
Formula
- 2×2: λ² − tr(A)λ + det(A) (up to leading coefficient convention)
- Diagonal 3×3: product of (d<sub>i</sub> − λ)
Introduction
Examples turn definitions into muscle memory. Each setup below shows the matrix, the expansion idea, and the factored form when reasonable. Use the Characteristic Polynomial Calculator to confirm arithmetic after you try the expansion yourself.
Start with 2×2 because quadratics factor quickly. Move to 3×3 when you can build cofactor tables without sign drift.
Applied interpretations are included so the polynomial connects to eigenvalues and simple dynamic models, not only symbolic manipulation.
Patterns to recognize before expanding
Symmetric 2×2 matrices often have real eigenvalues. The polynomial may factor into linear terms with rational roots.
Diagonal and triangular matrices produce product forms along the diagonal. Expand only when the assignment requires standard form.
Rotation blocks such as [[0, 1], [−1, 0]] produce p(λ) = λ² + 1, a standard example of complex eigenvalues from a real matrix.
Before studying roots in depth, know the manual expansion checklist so examples do not feel like memorized answers.
Engineering setups often begin with a small state matrix. Signs of real parts of eigenvalues (from roots of p(λ)) inform stability sketches in differential equations courses.
After you factor p(λ), the eigenvalue connection guide explains multiplicity and eigenvectors in more detail.
Formulas reused in every example
- p(λ) = det(A − λI)
- 2×2 shortcut after expansion
- p(λ) = 0 for eigenvalues
Write the formula at the top of each example before numbers appear. It keeps the task anchored.
When factoring fails over ℝ, state complex roots clearly. They are valid outcomes.
Step-by-step overview
- Classify the matrix structure. Diagonal, triangular, symmetric, or general? Structure picks the fastest path.
- Build A − λI. Subtract λ on the diagonal only.
- Expand or multiply factors. Use cofactors for general matrices; multiply diagonal factors when allowed.
- Factor p(λ) if feasible. Integer coefficients often allow rational root tests.
- State eigenvalues with multiplicity. Roots of p(λ) are eigenvalues over ℂ.
Worked 2×2 and 3×3 examples
Example 1: A = [[4, 1], [2, 3]]. Expansion gives p(λ) = λ² − 7λ + 10 = (λ − 2)(λ − 5). Eigenvalues 2 and 5.
Example 2: A = [[0, 1], [−1, 0]]. Here p(λ) = λ² + 1 with roots ±i.
Example 3: A = diag(1, −2, 3) gives p(λ) = (1−λ)(−2−λ)(3−λ). Expand if required.
Example 4 (ODE motivation): a 2×2 companion matrix from a second-order linear equation produces a p(λ) whose roots match the exponents in the solution basis.