Chapter 10: Chapter 10

Problem 242

Find the eigenvalues and the eigenvectors of the matrix \(\mathrm{A}\), where

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The eigenvalues λ₁ and λ₂ are the solutions of the quadratic equation \(λ^2 - λ(a+d) + (ad - bc) = 0\). For each eigenvalue, find the corresponding eigenvectors by solving the system of linear equations \((A - λI)X = 0\), where A is the given matrix and I is the identity matrix. The eigenvectors can be obtained using row-reduction or any other preferred method.

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TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Compute the determinant of (A - λI)

Given the matrix A, we first need to compute the determinant of the matrix (A - λI), where λ is the eigenvalue and I is the identity matrix. Let A = \[ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \] To obtain (A- λI), we subtract λ from the diagonal elements: \[ \begin{bmatrix} a-λ & b \\ c & d-λ \end{bmatrix} \] Now, find the determinant |A - λI|: \(|A - λI| = (a - λ)(d - λ) - bc\)

Step 2: Solve for λ

Now we need to set the determinant equation equal to zero and solve for λ: \((a - λ)(d - λ) - bc = 0\) By expanding the equation, we get a quadratic equation in terms of λ: \(λ^2 - λ(a+d) + (ad - bc) = 0\) Solve the quadratic equation to obtain the eigenvalues λ₁ and λ₂.

Step 3: Find the eigenvectors

Now, for each eigenvalue, λ, we need to find the eigenvectors that satisfy the equation (\(A - λI)X = 0\), where X is the eigenvector. For λ₁: 1. Substitute λ₁ into the matrix (A-λI): \[ \begin{bmatrix} a-λ₁ & b \\ c & d-λ₁ \end{bmatrix} \] 2. Solve the system of linear equations (\(A-λ₁I)X = 0\) either by row-reduction or any other preferred method: - If only one equation is left, then we have a one-parameter solution (or more), and we can parameterize the eigenvector X. - If two independent equations are left, then there are no eigenvectors for this eigenvalue. Repeat the same process for λ₂. Now, you have found the eigenvalues (λ₁, λ₂) and their corresponding eigenvectors.

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Chapter 10: Chapter 10

Find the eigenvalues and the eigenvectors of the matrix \(\mathrm{A}\), where

Short Answer

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Step 1: Compute the determinant of (A - λI)

Step 2: Solve for λ

Step 3: Find the eigenvectors

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Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Chapter 18

Chapter 19

Chapter 20

Chapter 21

Chapter 22

Chapter 23

Chapter 24

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