---
title: "Modal matrix of a matrix [1,0,−1;1,2,1;…]"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:modal-matrix-of-a-matrix-1"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/modal-matrix-of-a-matrix-1
site: BCT Engineering
---

# Modal matrix of a matrix [1,0,−1;1,2,1;…]

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2082 Shrawan, 5 marks; 2081 Baishakh, 5 marks (asked 2 times)
- Video: [Find Eigenvalues & Construct Modal Matrix | Matrix Diagonalization](https://www.youtube.com/watch?v=VLz0kONYkAo) — Mathematics Tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/modal-matrix-of-a-matrix-1
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Find the modal matrix of the matrix A = [1 0 −1; 1 2 1; 2 2 3].

## Short answer

The modal matrix is P = [[1, −2, −1], [−1, 1, 1], [0, 2, 2]] with columns as independent eigenvectors for λ=1,2,3; P⁻¹AP = diag(1,2,3).

## Answer

The modal matrix P of A is the matrix whose columns are the linearly independent eigenvectors of A; then P⁻¹AP = D, the diagonal matrix of eigenvalues.

A = [1 0 −1; 1 2 1; 2 2 3]

#### Eigenvalues
The characteristic equation is λ³ − 6λ² + 11λ − 6 = 0 (trace 6, sum of principal minors 11, |A| = 6), i.e. (λ − 1)(λ − 2)(λ − 3) = 0.

**λ = 1, 2, 3.**

#### Eigenvectors, from (A − λI)X = 0
**λ = 1:** A − I = [0 0 −1; 1 1 1; 2 2 2]. From R₁: z = 0; from R₂: x + y = 0.
X₁ = (1, −1, 0)ᵀ.

**λ = 2:** A − 2I = [−1 0 −1; 1 0 1; 2 2 1]. x + z = 0 and 2x + 2y + z = 0. Take z = 2: x = −2, y = 1.
X₂ = (−2, 1, 2)ᵀ.

**λ = 3:** A − 3I = [−2 0 −1; 1 −1 1; 2 2 0]. z = −2x and x + y = 0. Take x = −1: y = 1, z = 2.
X₃ = (−1, 1, 2)ᵀ.

#### Modal matrix
```text
P = [ 1 −2 −1]
    [−1  1  1]
    [ 0  2  2]      |P| = −2 ≠ 0
```

Since |P| ≠ 0 the eigenvectors are independent, and

```text
P⁻¹AP = D = [1 0 0]
            [0 2 0]
            [0 0 3]
```

**Answer: the modal matrix is P above (columns X₁, X₂, X₃), and the diagonal form is diag(1, 2, 3).** Any non-zero scalar multiples of the columns are also valid.

## Check yourself

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

1. What is the modal matrix of A?
   A. The matrix of eigenvalues
   B. The inverse of A
   C. The matrix whose columns are independent eigenvectors
   D. The matrix of cofactors
   Answer: C. Columns of P are eigenvectors and P^-1 A P = D. (Key point: Modal matrix)

2. Which are the eigenvalues?
   A. 1, 1, 6
   B. 1, 2, 3
   C. 0, 2, 4
   D. 2, 3, 6
   Answer: B. Characteristic equation is (l-1)(l-2)(l-3) = 0. (Key point: Eigenvalues)

3. The eigenvector for lambda = 1 satisfies
   A. z = 0 and x + y = 0, so (1, -1, 0)
   B. x = 0 and y + z = 0
   C. z = 0 and x = y
   D. x + y + z = 0 only, so (1,1,-2)
   Answer: A. A - I has rows (0 0 -1),(1 1 1). (Key point: Eigenvector for 1)

4. What is P^-1 A P?
   A. The identity matrix
   B. diag(6, 11, 6)
   C. P itself
   D. diag(1, 2, 3), valid because |P| != 0
   Answer: D. Eigenvalues appear in the same order as the eigenvector columns. (Key point: Diagonal form)

## Related in Mathematics III

- [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×
- [Modal matrix of a matrix [5,4;1,2]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Amodal-matrix-of-a-matrix-5) — Mathematics III, chapter 1, asked 2×
- [Modal matrix of a matrix [2,1,1;−2,1,3;…]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Amodal-matrix-of-a-matrix-2) — Mathematics III, chapter 1