2. Without using Sage, find • the characteristic polynomial the eigenvalues (include the multiplicity of each) bases for the eigenspaces (indicate the dimension of the eigenspaces) of the following matrices: (a) B= (b) C = 10 (c) D= -2 10 (Once you've found the characteristic polynomial, you may find -201 its factorization by using Sage.
2. Without using Sage, find • the characteristic polynomial the eigenvalues (include the multiplicity of each) bases for the eigenspaces (indicate the dimension of the eigenspaces) of the following matrices: (a) B= (b) C = 10 (c) D= -2 10 (Once you've found the characteristic polynomial, you may find -201 its factorization by using Sage.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Exercise 2: Eigenvalues and Eigenvectors without Sage**
For the matrices provided, perform the following tasks:
- Find the characteristic polynomial.
- Determine the eigenvalues, including their multiplicities.
- Identify bases for the eigenspaces, and indicate the dimension of each eigenspace.
**Matrices:**
(a)
\[
B = \begin{bmatrix}
3 & 0 \\
8 & -1
\end{bmatrix}
\]
(b)
\[
C = \begin{bmatrix}
10 & -9 \\
4 & -2
\end{bmatrix}
\]
(c)
\[
D = \begin{bmatrix}
4 & 0 & 1 \\
-2 & 1 & 0 \\
-2 & 0 & 1
\end{bmatrix}
\]
*Note:* After finding the characteristic polynomial, you can use Sage for factorization.
**Example of Using Sage:**
To factor \( x^2 - 3x + 2 \) in Sage, you can use the following code:
```python
f = x^2 - 3*x + 2
f.factor()
```](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F238fed3b-2fcf-40ca-a8e1-295985b4dcab%2Fa9620e3f-5d6b-44a2-9ed5-fb587fc4e410%2F2ii0mh_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise 2: Eigenvalues and Eigenvectors without Sage**
For the matrices provided, perform the following tasks:
- Find the characteristic polynomial.
- Determine the eigenvalues, including their multiplicities.
- Identify bases for the eigenspaces, and indicate the dimension of each eigenspace.
**Matrices:**
(a)
\[
B = \begin{bmatrix}
3 & 0 \\
8 & -1
\end{bmatrix}
\]
(b)
\[
C = \begin{bmatrix}
10 & -9 \\
4 & -2
\end{bmatrix}
\]
(c)
\[
D = \begin{bmatrix}
4 & 0 & 1 \\
-2 & 1 & 0 \\
-2 & 0 & 1
\end{bmatrix}
\]
*Note:* After finding the characteristic polynomial, you can use Sage for factorization.
**Example of Using Sage:**
To factor \( x^2 - 3x + 2 \) in Sage, you can use the following code:
```python
f = x^2 - 3*x + 2
f.factor()
```
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