---
title: "Cayley-Hamilton theorem verification [1,0,2;0,2,1;…]"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:cayley-hamilton-theorem-verification-1-0-two"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/cayley-hamilton-theorem-verification-1-0-two
site: BCT Engineering
---

# Cayley-Hamilton theorem verification [1,0,2;0,2,1;…]

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2080 Bhadra, 1+4 marks
- Video: [Verify Cayley Hamilton for the matrix & find its inverse best example(PART-2)](https://www.youtube.com/watch?v=qLOaCyqSuVA) — EASY MATHS EASY TRICKS
- 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-1-0-two
- 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 = [1 0 2; 0 2 1; 2 0 3].

## Short answer

Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. For A, the characteristic equation is λ³ − 6λ² + 7λ + 2 = 0. Computing A³ − 6A² + 7A + 2I gives the zero matrix, confirming 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 = [1 0 2; 0 2 1; 2 0 3]

#### Characteristic equation
- Trace = 1 + 2 + 3 = 6
- Sum of principal minors = (2·3 − 1·0) + (1·3 − 2·2) + (1·2 − 0) = 6 − 1 + 2 = 7
- |A| = 1(6 − 0) − 0 + 2(0 − 4) = 6 − 8 = −2

```text
λ³ − 6λ² + 7λ + 2 = 0
```

#### Verification: A³ − 6A² + 7A + 2I = O
```text
A² = [5 0  8]         A³ = A²·A = [21 0 34]
     [2 4  5]                      [12 8 23]
     [8 0 13]                      [34 0 55]
```

Entry by entry:

```text
Row 1: 21 − 30 + 7 + 2 = 0 ;  0 ;  34 − 48 + 14 = 0
Row 2: 12 − 12 + 0 = 0 ;  8 − 24 + 14 + 2 = 0 ;  23 − 30 + 7 = 0
Row 3: 34 − 48 + 14 = 0 ;  0 ;  55 − 78 + 21 + 2 = 0
```

All entries vanish, so A³ − 6A² + 7A + 2I = O. The theorem is verified. ∎

## Check yourself

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

1. What does the Cayley-Hamilton theorem say?
   A. Every square matrix satisfies its own characteristic equation
   B. Every matrix is diagonalizable
   C. The trace equals the determinant
   D. Every matrix has real eigenvalues
   Answer: A. If p(lambda) = 0 is the characteristic equation then p(A) = O. (Key point: Statement)

2. For a 3x3 matrix, the characteristic equation is
   A. lambda^3 + (trace) lambda^2 + |A| = 0
   B. lambda^3 - (trace) lambda^2 + (sum of principal minors) lambda - |A| = 0
   C. lambda^3 - |A| lambda^2 + trace = 0
   D. lambda^2 - trace lambda + |A| = 0
   Answer: B. Coefficients are trace, sum of principal 2x2 minors and the determinant, with alternating signs. (Key point: Building the characteristic equation)

3. For trace 6, minors sum 7 and |A| = -2, the equation is
   A. lambda^3 - 6 lambda^2 + 7 lambda + 2 = 0
   B. lambda^3 - 6 lambda^2 + 7 lambda - 2 = 0
   C. lambda^3 + 6 lambda^2 + 7 lambda + 2 = 0
   D. lambda^3 - 7 lambda^2 + 6 lambda + 2 = 0
   Answer: A. The constant is -|A| = +2. (Key point: Characteristic equation)

4. What matrix identity is verified?
   A. A^3 - 6A^2 + 7A - 2I = O
   B. A^3 - 6A^2 + 7A + 2I = O
   C. A^3 + 6A^2 + 7A + 2I = O
   D. A^3 - 6A^2 + 7A = 2I
   Answer: B. All entries of this combination vanish. (Key point: Verification)

## Close variants set in other exams

- [Cayley-Hamilton theorem verification [1,0,−1;1,2,1;…]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Acayley-hamilton-theorem-verification-1-0) — Mathematics III, chapter 1, asked 6×
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