---
title: "Cayley-Hamilton theorem verification [1,3,7;4,2,3;…]"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:cayley-hamilton-theorem-verification-1-3"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/cayley-hamilton-theorem-verification-1-3
site: BCT Engineering
---

# Cayley-Hamilton theorem verification [1,3,7;4,2,3;…]

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2078 Kartik, 5 marks; 2069 Asar, 5 marks (asked 2 times)
- 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-1-3
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

State Cayley-Hamilton theorem and verify the theorem for the square matrix A = [1 3 7; 4 2 3; 1 2 1].

## Short answer

Cayley-Hamilton theorem: every square matrix satisfies its own characteristic equation. For A, the characteristic polynomial is λ³ − 4λ² − 20λ − 35, and A³ − 4A² − 20A − 35I = 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 = [1 3 7; 4 2 3; 1 2 1]

#### Characteristic equation
- Trace = 1 + 2 + 1 = 4
- Sum of principal minors = (2·1 − 3·2) + (1·1 − 7·1) + (1·2 − 3·4) = −4 − 6 − 10 = −20
- |A| = 1(2 − 6) − 3(4 − 3) + 7(8 − 2) = −4 − 3 + 42 = 35

```text
λ³ − 4λ² − 20λ − 35 = 0
```

#### Verification: A³ − 4A² − 20A − 35I = O
```text
A² = [20 23 23]        A³ = A²·A = [135 152 232]
     [15 22 37]                     [140 163 208]
     [10  9 14]                     [ 60  76 111]
```

Entry by entry:

```text
Row 1: 135 − 80 − 20 − 35 = 0 ;  152 − 92 − 60 = 0 ;  232 − 92 − 140 = 0
Row 2: 140 − 60 − 80 = 0 ;  163 − 88 − 40 − 35 = 0 ;  208 − 148 − 60 = 0
Row 3:  60 − 40 − 20 = 0 ;   76 − 36 − 40 = 0 ;  111 − 56 − 20 − 35 = 0
```

All entries are zero, so A³ − 4A² − 20A − 35I = 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 4, principal minors sum -20 and |A| = 35, the equation is
   A. lambda^3 - 4 lambda^2 + 20 lambda - 35 = 0
   B. lambda^3 + 4 lambda^2 - 20 lambda + 35 = 0
   C. lambda^3 - 4 lambda^2 - 20 lambda + 35 = 0
   D. lambda^3 - 4 lambda^2 - 20 lambda - 35 = 0
   Answer: D. -trace = -4, +(-20) = -20, -|A| = -35. (Key point: Characteristic equation)

4. In the matrix identity the constant term is
   A. -35 I
   B. -35
   C. 35 A
   D. -35 A^0 with A^0 = 0
   Answer: A. The scalar is multiplied by the identity matrix. (Key point: Constant term)

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

## Related in Mathematics III

- [Cayley-Hamilton theorem verification [2,−1,1;−1,2,−1;…]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Acayley-hamilton-theorem-verification-2-1) — Mathematics III, chapter 1
- [Cayley-Hamilton theorem verification [1,1,3;1,3,−3;…]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Acayley-hamilton-theorem-verification-1-1) — Mathematics III, chapter 1, asked 3×
- [Cayley-Hamilton theorem verification [4,3,1;2,1,−2;…]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Acayley-hamilton-theorem-verification-4-3) — Mathematics III, chapter 1, asked 3×
- [Cayley-Hamilton theorem verification [−2,2,−3;2,1,−6;…]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Acayley-hamilton-theorem-verification-2-2-two) — Mathematics III, chapter 1, asked 3×