---
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"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/eigenvalues-and-eigenvectors-2-2-2
site: BCT Engineering
---

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

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2068 Baishakh, 5 marks; 2067 Magh (asked 2 times)
- 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
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Find the eigenvalues and eigenvectors of the matrix [2 −2 2; 1 1 1; 1 3 −1].

## Short answer

Eigenvalues: λ = −2, 2 (double). Eigenvectors: for λ=−2, (4,1,−7)ᵀ; for λ=2, (0,1,1)ᵀ (only one independent vector).

## Answer

Eigenvalues are the roots of |A − λI| = 0, and each eigenvector solves (A − λI)X = 0.

A = [2 −2 2; 1 1 1; 1 3 −1]

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

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

**Eigenvalues: λ = 2, 2, −2.**

#### Eigenvector for λ = −2
```text
A + 2I = [4 −2  2]
         [1  3  1]
         [1  3  1]
```
Equations: x + 3y + z = 0 and 4x − 2y + 2z = 0 (i.e. z = y − 2x). Substituting: x + 3y + y − 2x = 0 ⇒ x = 4y, then z = −7y.

X₁ = (4, 1, −7)ᵀ.

#### Eigenvector for λ = 2
```text
A − 2I = [0 −2  2]
         [1 −1  1]
         [1  3 −3]
```
From R₁: y = z. From R₂: x − y + z = 0 ⇒ x = 0. R₃ is then satisfied (3y − 3z = 0). The rank of A − 2I is 2, so only one independent vector exists:

X₂ = (0, 1, 1)ᵀ.

**Answer:**
| Eigenvalue | Eigenvector |
|---|---|
| −2 | k(4, 1, −7)ᵀ |
| 2 (twice) | k(0, 1, 1)ᵀ |

The repeated root 2 has only one independent eigenvector, so A is not diagonalizable.

## Check yourself

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

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

2. Which is an eigenvector for lambda = -2?
   A. (4, 1, -7)
   B. (0, 1, 1)
   C. (1, 1, 1)
   D. (4, 1, 7)
   Answer: A. x = 4y and z = -7y. (Key point: Eigenvector for -2)

3. How many independent eigenvectors does lambda = 2 have?
   A. 2
   B. 3
   C. 1, since rank(A - 2I) = 2
   D. 0
   Answer: C. n - rank = 3 - 2 = 1. (Key point: Repeated root)

4. Is A diagonalizable?
   A. Yes, since eigenvalues are real
   B. Yes, since trace is 2
   C. No, since |A| = 0
   D. No, only two independent eigenvectors in total
   Answer: D. Repeated eigenvalue 2 lacks a second independent eigenvector. (Key point: Diagonalizability)

## Close variants set in other exams

- [Eigenvalues and eigenvectors [2,2,1;1,3,1;…]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aeigenvalues-and-eigenvectors-2-2-1) — Mathematics III, chapter 1, asked 8×
- [Eigenvalues and eigenvectors [1,2,−1;0,2,1;…]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aeigenvalues-and-eigenvectors-1-2-1) — Mathematics III, chapter 1, asked 8×
- [Eigenvalues and eigenvectors [−2,2,−3;2,1,−6;…]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aeigenvalues-and-eigenvectors-2-2-3-two) — Mathematics III, chapter 1, asked 8×
- [Eigenvalues and eigenvectors [2,1,1;−2,1,3;…]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aeigenvalues-and-eigenvectors-2-1-1-two) — Mathematics III, chapter 1, asked 8×
- [Eigenvalues, eigenvectors, modal matrix [2,0,1;0,2,−1;…]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aeigenvalues-eigenvectors-modal-matrix-2-0) — Mathematics III, chapter 1, asked 8×

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