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

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

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2073 Shrawan, 4+1 marks
- Video: [3 x  3 eigenvalues and eigenvectors](https://www.youtube.com/watch?v=qa9fI6qvUQY) — Prime Newtons
- 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-2-1-1
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Find the eigenvalues and eigenvectors of the matrix [2 1 1; 1 2 1; 0 0 1]. Give an example showing the importance of eigenvectors.

## Short answer

Eigenvalues: 1, 1, 3. Eigenvectors: for λ=3, k(1,1,0); for λ=1, any combination of (−1,1,0) and (−1,0,1). They allow diagonalization to compute Aⁿ easily, e.g., Aⁿ = P diag(1,1,3ⁿ) P⁻¹.

## Answer

A = [2 1 1; 1 2 1; 0 0 1]

#### Eigenvalues
Expanding |A − λI| along the third row:

```text
|A − λI| = (1 − λ) · | 2−λ  1  |
                     |  1  2−λ | = (1 − λ)[(2 − λ)² − 1] = (1 − λ)(1 − λ)(3 − λ)
```

**Eigenvalues: λ = 1, 1, 3.**

#### Eigenvectors
**λ = 3:** A − 3I = [−1 1 1; 1 −1 1; 0 0 −2]. From R₃: z = 0; then x = y.
X₁ = (1, 1, 0)ᵀ.

**λ = 1:** A − I = [1 1 1; 1 1 1; 0 0 0]. The single equation x + y + z = 0 (rank 1) has two independent solutions:
- X₂ = (−1, 1, 0)ᵀ
- X₃ = (−1, 0, 1)ᵀ

| Eigenvalue | Eigenvectors |
|---|---|
| 3 | k(1, 1, 0)ᵀ |
| 1 (twice) | k₁(−1, 1, 0)ᵀ + k₂(−1, 0, 1)ᵀ |

#### Example showing the importance of eigenvectors
Eigenvectors let us **diagonalize** a matrix and compute its powers easily. Take P = [X₂ X₃ X₁] = [−1 −1 1; 1 0 1; 0 1 0] and D = diag(1, 1, 3). Then A = PDP⁻¹ and

```text
Aⁿ = P Dⁿ P⁻¹ = P diag(1, 1, 3ⁿ) P⁻¹
   = [(3ⁿ+1)/2  (3ⁿ−1)/2  (3ⁿ−1)/2]
     [(3ⁿ−1)/2  (3ⁿ+1)/2  (3ⁿ−1)/2]
     [   0          0          1    ]
```

For n = 3 this gives A³ = [14 13 13; 13 14 13; 0 0 1], which agrees with multiplying A three times. Without eigenvectors we would have to multiply the matrix n times. The same idea is used to solve systems of differential equations X′ = AX, to find principal axes of a quadratic form, and to study vibrations and stability.

## Check yourself

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

1. What are the eigenvalues?
   A. 1, 2, 3
   B. 1, 1, 3
   C. 1, 3, 3
   D. 2, 2, 1
   Answer: B. |A - λI| = (1-λ)^2(3-λ). (Key point: Eigenvalues)

2. The eigenvector for λ = 3 is:
   A. (1, -1, 0)
   B. (1, 1, 1)
   C. (1, 1, 0)
   D. (0, 0, 1)
   Answer: C. R3 gives z = 0 and then x = y. (Key point: Eigenvector for 3)

3. For λ = 1, A - I has rank 1. The number of independent eigenvectors is:
   A. 1
   B. 3
   C. 0
   D. 2
   Answer: D. 3 - 1 = 2, so A is diagonalizable. (Key point: Repeated eigenvalue)

4. The example shows eigenvectors are useful because:
   A. A^n = P D^n P^-1 gives powers of A without repeated multiplication
   B. They make A symmetric
   C. They give the determinant
   D. They remove the need for eigenvalues
   Answer: A. Diagonalizing means only the diagonal entries are raised to the power n. (Key point: Importance of eigenvectors)

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