---
title: "Determinant identity proof"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:determinant-identity-proof-four"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/determinant-identity-proof-four
site: BCT Engineering
---

# Determinant identity proof

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2078 Bhadra, 2+1+1+1 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/determinant-identity-proof-four
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Find the eigenvalues of the matrix A = [1 0 −1; 1 2 1; 2 2 3] and use them to compute (i) eigenvalues of A⁻¹ (ii) determinant of A (iii) eigenvalues of adj A.

## Short answer

Characteristic equation λ³ − 6λ² + 11λ − 6 = 0 gives eigenvalues 1, 2, 3. Determinant = 1·2·3 = 6. Eigenvalues of A⁻¹: 1, 1/2, 1/3; eigenvalues of adj A = |A|/λ: 6, 3, 2.

## Answer

A = [1 0 −1; 1 2 1; 2 2 3]

#### Eigenvalues
- Trace = 1 + 2 + 3 = 6
- Sum of principal minors = (6 − 2) + (3 + 2) + (2 − 0) = 4 + 5 + 2 = 11
- |A| = 1(6 − 2) − 0 + (−1)(2 − 4) = 4 + 2 = 6

```text
λ³ − 6λ² + 11λ − 6 = 0  ⇒  (λ − 1)(λ − 2)(λ − 3) = 0
```
**Eigenvalues of A: 1, 2, 3.**

#### (i) Eigenvalues of A⁻¹
If λ is an eigenvalue of a non-singular A, then 1/λ is an eigenvalue of A⁻¹.

**Eigenvalues of A⁻¹: 1, 1/2, 1/3.**

#### (ii) Determinant of A
The determinant is the product of the eigenvalues:

|A| = 1 × 2 × 3 = **6**

(This agrees with direct expansion.)

#### (iii) Eigenvalues of adj A
Since adj A = |A| A⁻¹, its eigenvalues are |A|/λ:

**Eigenvalues of adj A: 6/1, 6/2, 6/3 = 6, 3, 2.**

## Check yourself

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

1. With trace 6, sum of principal minors 11 and |A| = 6, what are the eigenvalues?
   A. 1, 2, 3
   B. 1, 1, 4
   C. 2, 2, 2
   D. -1, -2, -3
   Answer: A. λ^3 - 6λ^2 + 11λ - 6 = (λ-1)(λ-2)(λ-3). (Key point: Characteristic polynomial)

2. What are the eigenvalues of A^-1?
   A. -1, -2, -3
   B. 1, 2, 3
   C. 1, 4, 9
   D. 1, 1/2, 1/3
   Answer: D. If λ is an eigenvalue of non-singular A, 1/λ is an eigenvalue of A^-1. (Key point: Eigenvalues of inverse)

3. How is |A| obtained from the eigenvalues?
   A. Their sum
   B. Their product
   C. Their largest value
   D. Their sum of squares
   Answer: B. The determinant is the product of the eigenvalues: 1 x 2 x 3 = 6. (Key point: Determinant from eigenvalues)

4. Since adj A = |A| A^-1, what are the eigenvalues of adj A?
   A. 1, 2, 3
   B. 1/6, 1/3, 1/2
   C. 6, 3, 2
   D. 6, 12, 18
   Answer: C. Each eigenvalue is |A|/λ = 6/1, 6/2, 6/3. (Key point: Eigenvalues of adj A)

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