Quick answer

Build A − λI, compute p(λ) = det(A − λI), simplify to standard form, then solve p(λ) = 0 when eigenvalues are required.

Formula

  • A − λI
  • p(λ) = det(A − λI)
  • p(λ) = 0 → eigenvalues

Introduction

Students lose points when they skip determinant rules while rushing to eigenvalues. The method below keeps the order explicit: polynomial first, roots second. Use the Characteristic Polynomial Calculator to verify expansions after each manual attempt.

You should know what p(λ) represents before following the checklist. If vocabulary is shaky, read the definition guide first, then return here.

The steps scale from 2×2 drills to 4×4 assignments where arithmetic length is the main challenge.

What the step-by-step method accomplishes

The method turns a matrix into a polynomial in λ without numeric substitution. Each step has a clear output: labeled matrix, cofactor table, expanded determinant, simplified p(λ).

Constructing (A − λI) is the most common early mistake. Subtract λ only on the diagonal. Copy every off-diagonal entry exactly from A.

Computing determinants with λ symbols requires the same sign patterns as numeric determinants. A checkerboard for cofactor signs prevents systematic errors.

Polynomial simplification is not optional. Like terms in λ must be collected before you compare to the formula sheet or the calculator.

Eigenvalue extraction belongs after p(λ) is correct. Treat solving p(λ) = 0 as a separate phase so you do not blend expansion errors into root finding.

The calculator guide explains how to check expansions quickly when cofactor work becomes long.

Formulas used in each step

  • (A − λI)<sub>ii</sub> = a<sub>ii</sub> − λ
  • p(λ) = det(A − λI)
  • p(λ) = 0 defines eigenvalues

Write the formulas at the top of your scratch work before expanding. That keeps the task focused on execution rather than recall.

For 2×2, you may switch to λ² − tr(A)λ + det(A) after you prove the shortcut once on a generic matrix.

For 3×3, expect a mix of λ³, λ², and lower terms unless the matrix is triangular.

Step-by-step overview

  1. Label rows and columns. Use aij notation or row letters. Consistent labels reduce transpose-type errors.
  2. Build A − λI. Diagonal entries become aii − λ. Every other entry copies from A.
  3. Expand det(A − λI). Cofactor expansion along a row with structural zeros saves time on structured exam matrices.
  4. Simplify p(λ). Collect λn through λ0. Verify degree n.
  5. Sanity-check coefficients. Confirm p(0) = det(A). For 2×2, compare with trace and determinant shortcuts.
  6. Solve p(λ) = 0 if eigenvalues are requested. Factor, use quadratic/cubic methods, or approved numerical tools.

3×3 walkthrough sketch

Let A = diag(2, 3, 5). Then A − λI is diagonal with entries 2−λ, 3−λ, 5−λ, so p(λ) = (2−λ)(3−λ)(5−λ).

Expand the product to standard form if the problem requires expanded coefficients rather than factored form.

For a full non-diagonal 3×3, build a cofactor table and expand one row at a time. Do not skip writing minors because they keep signs organized.

Compare your final p(λ) with the home calculator before you move to eigenvectors.