Let -1 9 -1 1 1 2 0 |and P =| 0 0 1 1 0 5 A = 1 %3D 0 15 -2 Given that P diagonalizes A, then compute A". -1 -4104 1 -10245 1
Let -1 9 -1 1 1 2 0 |and P =| 0 0 1 1 0 5 A = 1 %3D 0 15 -2 Given that P diagonalizes A, then compute A". -1 -4104 1 -10245 1
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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Please, provide a different answer than the wong one that was provided twice now!
![Let
\[ A = \begin{bmatrix} -1 & 9 & -1 \\ 0 & 1 & 0 \\ 0 & 15 & -2 \end{bmatrix} \]
and
\[ P = \begin{bmatrix} 1 & 1 & 2 \\ 0 & 0 & 1 \\ 1 & 0 & 5 \end{bmatrix} \]
Given that \( P \) diagonalizes \( A \), then compute \( A^{11} \).
\[ A^{11} = \begin{bmatrix} -1 & -4104 & 2 \\ 0 & 1 & 0 \\ 0 & -10245 & 1 \end{bmatrix} \]
**Explanation:**
In the context of linear algebra, when a matrix \( A \) is diagonalizable, it means that there exists an invertible matrix \( P \) and a diagonal matrix \( D \) such that \( A = PDP^{-1} \). The power of a diagonalizable matrix \( A \) can be calculated efficiently using the relation \( A^n = PD^nP^{-1} \), where \( D^n \) is simply the diagonal matrix \( D \) raised to the power \( n \).
In this case, the matrix \( A^{11} \) has been computed utilizing the above property. The result shown is a matrix with elements resulting from several calculations which involve the powers of the diagonalized form of \( A \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9b53ebc-804c-49f3-b917-c4e1c4975b52%2F48d482df-6cdd-4fce-90bd-4c170bfa31c9%2Fcjymww3_processed.png&w=3840&q=75)
Transcribed Image Text:Let
\[ A = \begin{bmatrix} -1 & 9 & -1 \\ 0 & 1 & 0 \\ 0 & 15 & -2 \end{bmatrix} \]
and
\[ P = \begin{bmatrix} 1 & 1 & 2 \\ 0 & 0 & 1 \\ 1 & 0 & 5 \end{bmatrix} \]
Given that \( P \) diagonalizes \( A \), then compute \( A^{11} \).
\[ A^{11} = \begin{bmatrix} -1 & -4104 & 2 \\ 0 & 1 & 0 \\ 0 & -10245 & 1 \end{bmatrix} \]
**Explanation:**
In the context of linear algebra, when a matrix \( A \) is diagonalizable, it means that there exists an invertible matrix \( P \) and a diagonal matrix \( D \) such that \( A = PDP^{-1} \). The power of a diagonalizable matrix \( A \) can be calculated efficiently using the relation \( A^n = PD^nP^{-1} \), where \( D^n \) is simply the diagonal matrix \( D \) raised to the power \( n \).
In this case, the matrix \( A^{11} \) has been computed utilizing the above property. The result shown is a matrix with elements resulting from several calculations which involve the powers of the diagonalized form of \( A \).
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