Calculate M5 using a diagonalization process. =[63] has eigenvalues 5, 2. M =
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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![**Calculate \( M^5 \) using a diagonalization process.**
\[ M = \begin{bmatrix} 8 & 3 \\ -6 & -1 \end{bmatrix} \]
**Matrix \( M \) has eigenvalues 5, 2.**
To find \( M^5 \) using the diagonalization process, follow these steps:
1. **Find the Eigenvectors**: Use the eigenvalues 5 and 2 to find the corresponding eigenvectors.
2. **Form the Matrix \( P \)**: Construct the matrix \( P \) using the eigenvectors as columns.
3. **Diagonal Matrix \( D \)**: Create a diagonal matrix \( D \) with the eigenvalues on the diagonal:
\[ D = \begin{bmatrix} 5 & 0 \\ 0 & 2 \end{bmatrix} \]
4. **Compute \( D^5 \)**: Raise the diagonal matrix \( D \) to the power of 5:
\[ D^5 = \begin{bmatrix} 5^5 & 0 \\ 0 & 2^5 \end{bmatrix} = \begin{bmatrix} 3125 & 0 \\ 0 & 32 \end{bmatrix} \]
5. **Calculate \( M^5 \)**: Use the formula \( M^5 = P D^5 P^{-1} \) to get the result, where \( P^{-1} \) is the inverse of matrix \( P \).
These steps will give you the matrix \( M^5 \) using the diagonalization method.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0aa34ad3-f98c-40e8-8209-2950de228720%2F992bc657-b488-4552-bb90-9e264a6013dc%2Fbpvq6tm_processed.png&w=3840&q=75)
Transcribed Image Text:**Calculate \( M^5 \) using a diagonalization process.**
\[ M = \begin{bmatrix} 8 & 3 \\ -6 & -1 \end{bmatrix} \]
**Matrix \( M \) has eigenvalues 5, 2.**
To find \( M^5 \) using the diagonalization process, follow these steps:
1. **Find the Eigenvectors**: Use the eigenvalues 5 and 2 to find the corresponding eigenvectors.
2. **Form the Matrix \( P \)**: Construct the matrix \( P \) using the eigenvectors as columns.
3. **Diagonal Matrix \( D \)**: Create a diagonal matrix \( D \) with the eigenvalues on the diagonal:
\[ D = \begin{bmatrix} 5 & 0 \\ 0 & 2 \end{bmatrix} \]
4. **Compute \( D^5 \)**: Raise the diagonal matrix \( D \) to the power of 5:
\[ D^5 = \begin{bmatrix} 5^5 & 0 \\ 0 & 2^5 \end{bmatrix} = \begin{bmatrix} 3125 & 0 \\ 0 & 32 \end{bmatrix} \]
5. **Calculate \( M^5 \)**: Use the formula \( M^5 = P D^5 P^{-1} \) to get the result, where \( P^{-1} \) is the inverse of matrix \( P \).
These steps will give you the matrix \( M^5 \) using the diagonalization method.
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