Quick answer

If p(λ) = det(A − λI), then p(A) = 0 (zero matrix) using matrix arithmetic.

Formula

  • p(λ) = det(A − λI)
  • p(A) = 0
  • matrix powers in p(A)

Introduction

After you can compute p(λ), Cayley-Hamilton explains why that polynomial is not just a diagnostic tool: substituting A for λ yields zero. Use the Characteristic Polynomial Calculator to get p(λ), then verify p(A) on paper for small sizes.

The theorem connects polynomial algebra to matrix algebra in a concrete way.

Read this after characteristic and minimal polynomial comparisons make sense.

Theorem statement and matrix substitute

Theorem: let p(λ) = det(A − λI). Replace λ by the matrix A using matrix addition and multiplication. The result p(A) is the zero matrix.

Matrix substitute means λk becomes Ak, constants become scalar multiples of I, and coefficients multiply matrices on the left as usual.

Example intuition for 2×2: if p(λ) = λ² − tr(A)λ + det(A)I, then A² − tr(A)A + det(A)I = 0.

You need an accurate p(λ) before substitution. Use the formula guide or calculator, then substitute.

Applications include reducing high powers Ak, proving identities, and building toward Jordan form in advanced courses.

Compare with minimal polynomial, which may have smaller degree but still annihilates A.

What to verify on homework

  • p(λ) from det(A − λI)
  • p(A) using matrix operations
  • p(A) equals zero matrix entrywise

Instructors often ask for p(λ) first and Cayley-Hamilton verification second on the same matrix.

Keep dimensions consistent: I must match the size of A.

Step-by-step overview

  1. Find p(λ). Expansion or calculator.
  2. Write p(A). Replace λ with A; constants become cI.
  3. Compute powers needed. Usually A² for 2×2, more for larger n.
  4. Combine terms. Matrix addition term by term.
  5. Check zero matrix. Every entry should cancel.

2×2 verification

Let A = [[1, 1], [0, 2]]. Then p(λ) = (1−λ)(2−λ) = λ² − 3λ + 2.

Compute A² = [[1, 3], [0, 4]]. Then A² − 3A + 2I = [[0, 0], [0, 0]].

If a term fails to cancel, re-expand p(λ) before blaming the theorem.

Repeat on one 3×3 matrix only when your course requires it; arithmetic is longer but the logic is identical.