Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix. 60-3 17 3; λ=6, 7 00 7 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. For P = OB. For P = 600 D= 0 6 0 007 6 D = 0 7 0 007 C. The matrix cannot be diagonalized.
Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix. 60-3 17 3; λ=6, 7 00 7 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. For P = OB. For P = 600 D= 0 6 0 007 6 D = 0 7 0 007 C. The matrix cannot be diagonalized.
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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### Diagonalize the Matrix: Eigenvalues and Eigenvectors
**Problem Statement:**
Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix.
\[
\begin{bmatrix}
6 & 0 & -3 \\
1 & 7 & 3 \\
0 & 0 & 7
\end{bmatrix}
\]
\[ \lambda = 6, 7 \]
**Choices:**
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. For \( P = \, \_\_\_\_, \, D =
\begin{bmatrix}
6 & 0 & 0 \\
0 & 6 & 0 \\
0 & 0 & 7
\end{bmatrix}
\)
B. For \( P = \, \_\_\_\_, \, D =
\begin{bmatrix}
6 & 0 & 0 \\
0 & 7 & 0 \\
0 & 0 & 7
\end{bmatrix}
\)
C. The matrix cannot be diagonalized.
---
### Explanation:
1. **Matrix to be Diagonalized:**
\[
\begin{bmatrix}
6 & 0 & -3 \\
1 & 7 & 3 \\
0 & 0 & 7
\end{bmatrix}
\]
2. **Given Eigenvalues:**
- \(\lambda = 6, 7\)
3. **Choices for Diagonal Matrix \(D\) and Corresponding Matrix \(P\):**
- Choice A suggests a diagonal matrix \(D\) with eigenvalues 6, 6, and 7.
- Choice B suggests a diagonal matrix \(D\) with eigenvalues 6, 7, and 7.
- Choice C suggests that the given matrix cannot be diagonalized.
To solve this problem, identify the correct diagonal matrix \(D\) and corresponding matrix \(P\) that diagonalizes the given matrix.
Understanding how to determine the correct diagonal matrix involves finding the eigenvectors corresponding to the given eigenvalues, and testing the compatibility with given options.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff893ad80-e3fa-4d69-87b2-5d427e54a923%2F477088d9-e0d9-47b8-a83c-56021e782bfc%2Fh962k9c_processed.png&w=3840&q=75)
Transcribed Image Text:---
### Diagonalize the Matrix: Eigenvalues and Eigenvectors
**Problem Statement:**
Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix.
\[
\begin{bmatrix}
6 & 0 & -3 \\
1 & 7 & 3 \\
0 & 0 & 7
\end{bmatrix}
\]
\[ \lambda = 6, 7 \]
**Choices:**
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. For \( P = \, \_\_\_\_, \, D =
\begin{bmatrix}
6 & 0 & 0 \\
0 & 6 & 0 \\
0 & 0 & 7
\end{bmatrix}
\)
B. For \( P = \, \_\_\_\_, \, D =
\begin{bmatrix}
6 & 0 & 0 \\
0 & 7 & 0 \\
0 & 0 & 7
\end{bmatrix}
\)
C. The matrix cannot be diagonalized.
---
### Explanation:
1. **Matrix to be Diagonalized:**
\[
\begin{bmatrix}
6 & 0 & -3 \\
1 & 7 & 3 \\
0 & 0 & 7
\end{bmatrix}
\]
2. **Given Eigenvalues:**
- \(\lambda = 6, 7\)
3. **Choices for Diagonal Matrix \(D\) and Corresponding Matrix \(P\):**
- Choice A suggests a diagonal matrix \(D\) with eigenvalues 6, 6, and 7.
- Choice B suggests a diagonal matrix \(D\) with eigenvalues 6, 7, and 7.
- Choice C suggests that the given matrix cannot be diagonalized.
To solve this problem, identify the correct diagonal matrix \(D\) and corresponding matrix \(P\) that diagonalizes the given matrix.
Understanding how to determine the correct diagonal matrix involves finding the eigenvectors corresponding to the given eigenvalues, and testing the compatibility with given options.
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