---
title: "Cayley-Hamilton theorem verification [−2,2,−3;2,1,−6;…]"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:cayley-hamilton-theorem-verification-2-2"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/cayley-hamilton-theorem-verification-2-2
site: BCT Engineering
---

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

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2072 Chaitra, 1+4 marks
- Video: [Linear Algebra | Matrices | Cayley-Hamilton (CH) Theorem - Verification](https://www.youtube.com/watch?v=d2sQvjhhDnY) — The Math Virtuoso
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/cayley-hamilton-theorem-verification-2-2
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

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

## 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.

## Answer

**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

```text
λ³ + λ² − 21λ − 45 = 0
```

#### Verification: A³ + A² − 21A − 45I = O
```text
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):

```text
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. ∎

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. To verify Cayley-Hamilton you must show:
   A. A^3 = I
   B. The characteristic polynomial evaluated at A gives the zero matrix
   C. |A| = 0
   D. A^2 = A
   Answer: B. A must satisfy its own characteristic equation. (Key point: Statement)

2. For this A the characteristic equation is:
   A. λ^3 - λ^2 - 21λ + 45 = 0
   B. λ^3 + λ^2 + 21λ - 45 = 0
   C. λ^3 + λ^2 - 21λ - 45 = 0
   D. λ^3 - 21λ - 45 = 0
   Answer: C. Trace -1 gives +λ^2, minors -21, and |A| = 45 gives -45. (Key point: Characteristic equation)

3. In the verification A^3 + A^2 - 21A - 45I = O, what does the 45I term represent?
   A. The constant term multiplied by the identity
   B. The trace times A
   C. The determinant of A^2
   D. A scalar number
   Answer: A. A constant must be replaced by constant times I when a matrix is substituted. (Key point: Constant term)

4. What shows the theorem is verified?
   A. A^2 has all positive entries
   B. A^3 + A^2 - 21A - 45I has every entry zero
   C. |A| = 45
   D. A is invertible
   Answer: B. All entries of the combination come to 0. (Key point: Conclusion)

## Close variants set in other exams

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