6 0 Let A = |0 3 Lo 3 3] (a) Find the characteristic polynomial of A. (b) Find the eigenvalues of A and their corresponding eigenspaces. (c) Is A diagonalizable? Explain.
6 0 Let A = |0 3 Lo 3 3] (a) Find the characteristic polynomial of A. (b) Find the eigenvalues of A and their corresponding eigenspaces. (c) Is A diagonalizable? Explain.
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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![### Matrix Analysis
Given the matrix \( A \):
\[ A = \begin{bmatrix} 6 & 0 & 0 \\ 0 & 3 & 3 \\ 0 & 3 & 3 \end{bmatrix} \]
We are required to perform the following tasks:
#### (a) Find the characteristic polynomial of \( A \).
The characteristic polynomial of a matrix \( A \) is given by the determinant:
\[ \text{det}(A - \lambda I) \]
where \( \lambda \) is a scalar and \( I \) is the identity matrix. For the matrix given, we have:
\[ A - \lambda I = \begin{bmatrix} 6 - \lambda & 0 & 0 \\ 0 & 3 - \lambda & 3 \\ 0 & 3 & 3 - \lambda \end{bmatrix} \]
#### (b) Find the eigenvalues of \( A \) and their corresponding eigenspaces.
To find the eigenvalues, solve the characteristic equation obtained in part (a). The roots of this polynomial are the eigenvalues of \( A \). Once the eigenvalues are found, the eigenspaces can be determined by solving:
\[ (A - \lambda I) \mathbf{x} = 0 \]
for each eigenvalue \( \lambda \).
#### (c) Is \( A \) diagonalizable? Explain.
A matrix \( A \) is diagonalizable if there exists a matrix \( P \) such that \( P^{-1}AP \) is a diagonal matrix. This typically occurs if \( A \) has \( n \) linearly independent eigenvectors, where \( n \) is the size of the matrix.
1. **Characteristic Polynomial:**
\[
\text{det}(A - \lambda I) = \text{det}\left( \begin{bmatrix}
6 - \lambda & 0 & 0 \\
0 & 3 - \lambda & 3 \\
0 & 3 & 3 - \lambda
\end{bmatrix} \right)
\]
2. **Eigenvalues:**
Solve the characteristic polynomial to find the eigenvalues. Then, determine the eigenspaces for each eigenvalue \( \lambda \).
3. **Diagonalizability:**
Check if the number of linearly](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F15c5ec76-8bab-480b-9121-367c8408675d%2F5d8f5847-f95b-4db0-ab20-ca4ec0f889bb%2Fjn5nzpc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Matrix Analysis
Given the matrix \( A \):
\[ A = \begin{bmatrix} 6 & 0 & 0 \\ 0 & 3 & 3 \\ 0 & 3 & 3 \end{bmatrix} \]
We are required to perform the following tasks:
#### (a) Find the characteristic polynomial of \( A \).
The characteristic polynomial of a matrix \( A \) is given by the determinant:
\[ \text{det}(A - \lambda I) \]
where \( \lambda \) is a scalar and \( I \) is the identity matrix. For the matrix given, we have:
\[ A - \lambda I = \begin{bmatrix} 6 - \lambda & 0 & 0 \\ 0 & 3 - \lambda & 3 \\ 0 & 3 & 3 - \lambda \end{bmatrix} \]
#### (b) Find the eigenvalues of \( A \) and their corresponding eigenspaces.
To find the eigenvalues, solve the characteristic equation obtained in part (a). The roots of this polynomial are the eigenvalues of \( A \). Once the eigenvalues are found, the eigenspaces can be determined by solving:
\[ (A - \lambda I) \mathbf{x} = 0 \]
for each eigenvalue \( \lambda \).
#### (c) Is \( A \) diagonalizable? Explain.
A matrix \( A \) is diagonalizable if there exists a matrix \( P \) such that \( P^{-1}AP \) is a diagonal matrix. This typically occurs if \( A \) has \( n \) linearly independent eigenvectors, where \( n \) is the size of the matrix.
1. **Characteristic Polynomial:**
\[
\text{det}(A - \lambda I) = \text{det}\left( \begin{bmatrix}
6 - \lambda & 0 & 0 \\
0 & 3 - \lambda & 3 \\
0 & 3 & 3 - \lambda
\end{bmatrix} \right)
\]
2. **Eigenvalues:**
Solve the characteristic polynomial to find the eigenvalues. Then, determine the eigenspaces for each eigenvalue \( \lambda \).
3. **Diagonalizability:**
Check if the number of linearly
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