---
title: "Eigenvalues and eigenvectors [2,−2,2;1,1,1;…]"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:eigenvalues-and-eigenvectors-2-2-2-two"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/eigenvalues-and-eigenvectors-2-2-2-two
site: BCT Engineering
---

# Eigenvalues and eigenvectors [2,−2,2;1,1,1;…]

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2080 Baishakh, 3+1+1 marks
- Video: [Finding Eigenvalues and Eigenvectors](https://www.youtube.com/watch?v=TQvxWaQnrqI) — Professor Dave Explains
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/eigenvalues-and-eigenvectors-2-2-2-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Find the eigenvalues of the matrix A = [2 −2 2; 1 1 1; 1 3 −1]. Hence use them to compute a) |A|  b) the eigenvalues of A⁻¹.

## Short answer

The characteristic equation λ³ − 2λ² − 4λ + 8 = 0 yields eigenvalues 2, 2, −2. Determinant = product = −8. Eigenvalues of A⁻¹ are the reciprocals: 1/2, 1/2, −1/2.

## Answer

A = [2 −2 2; 1 1 1; 1 3 −1]

#### Eigenvalues
- Trace = 2
- Sum of principal minors = (−1 − 3) + (−2 − 2) + (2 + 2) = −4
- |A| = 2(−1 − 3) + 2(−1 − 1) + 2(3 − 1) = −8

```text
λ³ − 2λ² − 4λ + 8 = 0  ⇒  (λ − 2)²(λ + 2) = 0
```

**Eigenvalues: 2, 2, −2.**

#### (a) Determinant
The determinant equals the product of the eigenvalues:

|A| = 2 × 2 × (−2) = **−8**

(Direct expansion also gives −8, which confirms the result.)

#### (b) Eigenvalues of A⁻¹
If λ is an eigenvalue of a non-singular matrix A, then 1/λ is an eigenvalue of A⁻¹ (from AX = λX ⇒ X = λA⁻¹X ⇒ A⁻¹X = (1/λ)X). Since |A| ≠ 0, A⁻¹ exists:

**Eigenvalues of A⁻¹: 1/2, 1/2, −1/2.**

## Check yourself

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

1. The eigenvalues of A are
   A. 2, 2, -2
   B. 2, -2, -2
   C. -2, 4, 2
   D. 2, 2, 4
   Answer: A. (l-2)^2 (l+2) = 0. (Key point: Eigenvalues)

2. The determinant equals
   A. the sum of the eigenvalues, 2
   B. the product of the eigenvalues, -8
   C. the product of eigenvalues, 8
   D. the largest eigenvalue, 2
   Answer: B. 2 x 2 x (-2) = -8. (Key point: Determinant from eigenvalues)

3. The eigenvalues of A^-1 are
   A. 2, 2, -2
   B. -2, -2, 2
   C. 1/2, 1/2, -1/2
   D. 1/4, 1/4, 1/4
   Answer: C. If AX = lambda X then A^-1 X = (1/lambda) X. (Key point: Eigenvalues of inverse)

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