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Cayley-Hamilton theorem verification [−2,2,−3;2,1,−6;…]

Short answer

The Cayley-Hamilton theorem states a square matrix satisfies its characteristic equation. For A, the characteristic equation is λ³ + λ² − 21λ − 45 = 0; A³ + A² − 21A − 45I = O, verifying the theorem.

State Cayley-Hamilton theorem and verify it for the matrix A = [−2 2 −3; 2 1 −6; −1 −2 0].

Cayley–Hamilton theorem: every square matrix satisfies its own characteristic equation. If |A − λI| = λ³ + c₁λ² + c₂λ + c₃ = 0, then A³ + c₁A² + c₂A + c₃I = O.

A = [−2 2 −3; 2 1 −6; −1 −2 0]

Characteristic equation

  • Trace = −1
  • Sum of principal minors = (1·0 − (−6)(−2)) + ((−2)·0 − (−3)(−1)) + ((−2)(1) − 2·2) = −12 − 3 − 6 = −21
  • |A| = −2(0 − 12) − 2(0 − 6) + (−3)(−4 + 1) = 24 + 12 + 9 = 45
λ³ + λ² − 21λ − 45 = 0

Verification: A³ + A² − 21A − 45I = O

A² = [11   4  −6]        A³ = [ −8   38  −57]
     [ 4  17 −12]             [ 38   49 −114]
     [−2  −4  15]             [−19  −38   30]

Entry by entry (A³ + A² − 21A − 45I):

Row 1: −8 + 11 + 42 − 45 = 0 ;  38 + 4 − 42 = 0 ;  −57 − 6 + 63 = 0
Row 2: 38 + 4 − 42 = 0 ;  49 + 17 − 21 − 45 = 0 ;  −114 − 12 + 126 = 0
Row 3: −19 − 2 + 21 = 0 ;  −38 − 4 + 42 = 0 ;  30 + 15 − 0 − 45 = 0

All entries are zero, so the theorem is verified for A. ∎

✓ Answer checked by Claude Opus 5.5 (high effort)

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