List the eigenvalues of the matrix A₁, A₂= -1,1 Find an eigenvector ₁ for A₁ = -1: 1 3 Find an eigenvector ₂ for A₂ = 1 : 1 1 23 The fundamental matrix X(t) is ▼ 3 ▼ Using the eigenvectors ₁ and 2 that you found in parts 2 and 3, find the fundamental matrix solution = X(t)e to the system Par Part Part 4
List the eigenvalues of the matrix A₁, A₂= -1,1 Find an eigenvector ₁ for A₁ = -1: 1 3 Find an eigenvector ₂ for A₂ = 1 : 1 1 23 The fundamental matrix X(t) is ▼ 3 ▼ Using the eigenvectors ₁ and 2 that you found in parts 2 and 3, find the fundamental matrix solution = X(t)e to the system Par Part Part 4
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Linear Algebra: Eigenvalues and Eigenvectors**
**1. List the Eigenvalues of the Matrix**
Consider the matrix:
\[
\begin{bmatrix}
2 & -1 \\
3 & -2
\end{bmatrix}
\]
The eigenvalues, denoted as \( \lambda_1, \lambda_2 \), are:
\[
-1, 1
\]
---
**2. Find an Eigenvector \( \mathbf{v}_1 \) for \( \lambda_1 = -1 \):**
The eigenvector \( \mathbf{v}_1 \) associated with the eigenvalue \( \lambda_1 = -1 \) is:
\[
\begin{bmatrix}
1 \\
3
\end{bmatrix}
\]
---
**3. Find an Eigenvector \( \mathbf{v}_2 \) for \( \lambda_2 = 1 \):**
The eigenvector \( \mathbf{v}_2 \) associated with the eigenvalue \( \lambda_2 = 1 \) is:
\[
\begin{bmatrix}
1 \\
1
\end{bmatrix}
\]
---
**4. Using the Eigenvectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \), Find the Fundamental Matrix Solution**
Given the eigenvectors \(\mathbf{v}_1\) and \(\mathbf{v}_2\) found above, the task is to find the fundamental matrix solution for the system:
\[
\mathbf{\dot{x}} = \begin{bmatrix}
2 & -1 \\
3 & -2
\end{bmatrix} \mathbf{x}
\]
The fundamental matrix \( X(t) \) is calculated using these eigenvectors.
Note: The boxes in the original image should be filled with the appropriate fundamental solutions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4100c518-1a8f-4898-88a2-d0f1a7669694%2F04a53dec-2320-455d-8da1-8a8b976bdeb9%2Febfwa2i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Linear Algebra: Eigenvalues and Eigenvectors**
**1. List the Eigenvalues of the Matrix**
Consider the matrix:
\[
\begin{bmatrix}
2 & -1 \\
3 & -2
\end{bmatrix}
\]
The eigenvalues, denoted as \( \lambda_1, \lambda_2 \), are:
\[
-1, 1
\]
---
**2. Find an Eigenvector \( \mathbf{v}_1 \) for \( \lambda_1 = -1 \):**
The eigenvector \( \mathbf{v}_1 \) associated with the eigenvalue \( \lambda_1 = -1 \) is:
\[
\begin{bmatrix}
1 \\
3
\end{bmatrix}
\]
---
**3. Find an Eigenvector \( \mathbf{v}_2 \) for \( \lambda_2 = 1 \):**
The eigenvector \( \mathbf{v}_2 \) associated with the eigenvalue \( \lambda_2 = 1 \) is:
\[
\begin{bmatrix}
1 \\
1
\end{bmatrix}
\]
---
**4. Using the Eigenvectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \), Find the Fundamental Matrix Solution**
Given the eigenvectors \(\mathbf{v}_1\) and \(\mathbf{v}_2\) found above, the task is to find the fundamental matrix solution for the system:
\[
\mathbf{\dot{x}} = \begin{bmatrix}
2 & -1 \\
3 & -2
\end{bmatrix} \mathbf{x}
\]
The fundamental matrix \( X(t) \) is calculated using these eigenvectors.
Note: The boxes in the original image should be filled with the appropriate fundamental solutions.
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