---
title: "Eigenvalues and eigenvectors [2,0,1;0,2,−1;…]"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:eigenvalues-and-eigenvectors-2-0-1"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/eigenvalues-and-eigenvectors-2-0-1
site: BCT Engineering
---

# Eigenvalues and eigenvectors [2,0,1;0,2,−1;…]

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2082 Baishakh, 2+3 marks; 2070 Chaitra, 5 marks (asked 2 times)
- 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-2-0-1
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Find the eigenvalues and eigenvectors of [2 0 1; 0 2 −1; 0 0 2].

## Short answer

Eigenvalue 2 (algebraic multiplicity 3). Eigenvectors are (1,0,0)ᵀ and (0,1,0)ᵀ and their non-zero combinations; geometric multiplicity is 2, so the matrix is not diagonalizable.

## Answer

Eigenvalues are the roots of |A − λI| = 0; the eigenvector for λ solves (A − λI)X = 0.

A = [2 0 1; 0 2 −1; 0 0 2]

#### Eigenvalues
A is upper triangular, so its eigenvalues are the diagonal entries:

```text
|A − λI| = (2 − λ)³ = 0   ⇒   λ = 2, 2, 2
```

So λ = 2 has algebraic multiplicity 3.

#### Eigenvectors for λ = 2
```text
A − 2I = [0 0  1]
         [0 0 −1]
         [0 0  0]
```

Rank of (A − 2I) = 1. The equations reduce to z = 0, with x and y free. Number of independent vectors = 3 − 1 = 2.

Taking x = 1, y = 0 and x = 0, y = 1:

```text
X₁ = (1, 0, 0)ᵀ ,   X₂ = (0, 1, 0)ᵀ
```

The general eigenvector is X = (k₁, k₂, 0)ᵀ, with k₁, k₂ not both zero.

#### Remark
The algebraic multiplicity (3) exceeds the geometric multiplicity (2), so there are only two independent eigenvectors and A is **not diagonalizable**.

**Answer: eigenvalue 2 (repeated three times); eigenvectors (1, 0, 0)ᵀ and (0, 1, 0)ᵀ and their non-zero combinations.**

## Check yourself

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

1. The eigenvalues of an upper triangular matrix are
   A. its off-diagonal entries
   B. always distinct
   C. the row sums
   D. its diagonal entries
   Answer: D. The determinant (2-l)^3 gives l = 2 thrice. (Key point: Eigenvalues of triangular matrix)

2. What is the rank of A - 2I?
   A. 1
   B. 2
   C. 3
   D. 0
   Answer: A. Only the last column is non-zero and the two rows are proportional. (Key point: Rank of A - 2I)

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

4. Why is A not diagonalizable?
   A. Its eigenvalues are complex
   B. It is singular
   C. Algebraic multiplicity 3 exceeds geometric multiplicity 2
   D. It is not square
   Answer: C. Only 2 independent eigenvectors exist, fewer than 3. (Key point: Diagonalizability)

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