Quick answer

p(λ) = det(A − λI) has degree n and constant term det(A).

Formula

  • p(0) = det(A)
  • deg p = n
  • tr(A) links to λ^(n−1) coefficient

Introduction

If determinants are still shaky, characteristic polynomials will feel mysterious. This guide connects determinant properties to p(λ) and to singularity tests. Use the Characteristic Polynomial Calculator after manual attempts.

Trace and determinant invariants appear as coefficients and as eigenvalue summaries.

Read this alongside the formula guide for a complete coefficient story.

Determinant properties in λ expansions

Row swaps flip the sign of det(A − λI). Scaling a row scales the determinant. Adding a multiple of one row to another preserves the determinant.

Polynomial degree is n because each diagonal entry of A − λI is linear in λ and the determinant is multilinear in rows.

Matrix singularity: A is singular exactly when det(A) = 0, which is p(0) for the standard expanded form. Equivalently, λ = 0 is an eigenvalue.

Cofactor expansion is the manual engine behind many homework problems. The how-to-find guide applies those rules systematically.

Trace relationships appear in the λn−1 coefficient for det(A − λI). Use them as quick checks after expansion.

When singularity tests meet eigenvalue theory, read characteristic polynomial and eigenvalues for the vector viewpoint.

Invariant lines to remember

  • p(λ) = det(A − λI)
  • p(0) = det(A)
  • Product of eigenvalues relates to det(A) (with multiplicity, up to sign conventions)

Write p(0) = det(A) on every scratch page. It is the fastest coefficient check.

For 2×2, trace and determinant also match sum and product of eigenvalues in common conventions.

Step-by-step overview

  1. Review determinant rules. Keep a mini cheat sheet visible.
  2. Build A − λI. Diagonal linear terms drive degree n.
  3. Expand with cofactors. Track signs with a checkerboard.
  4. Read p(0). Equals det(A).
  5. Connect to eigenvalues. det(A) = 0 means zero is a root of p(λ).

Coefficient checks

After expanding p(λ) for a 2×2 matrix, verify p(0) equals the 2×2 determinant ad − bc.

For A = [[2, 0], [0, 3]], p(λ) = (2−λ)(3−λ) and det(A) = 6 matches the constant term in expanded form.

Singular A has det(A) = 0, so λ = 0 appears as a root of p(λ).

Use the calculator to cross-check constant terms when manual expansion is long.