Compute A5 if A = PDP-1. 1 0 0 2 1 4 P = 0 -1 3 D = 0 0 -1 00-1 A5 =
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![**Matrix Exponentiation Using Diagonalization**
In this example, we are asked to compute \( A^5 \) given that the matrix \( A \) can be expressed in the form \( A = PDP^{-1} \).
Here, we have:
\[ P = \begin{bmatrix} 1 & 4 & 1 \\ 0 & -1 & 3 \\ 0 & 0 & -1 \end{bmatrix} \]
\[ D = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1 \end{bmatrix} \]
**Matrix \( A^5 \)**
The power of matrix \( A \), specifically \( A^5 \), can be computed by using the property of diagonalization:
\[ A^5 = (PDP^{-1})^5 = PD^5P^{-1} \]
Here, \( D^5 \) involves raising the diagonal matrix \( D \) to the power of 5. Since \( D \) is diagonal, its powers can be computed by raising each of the diagonal elements to the power of 5:
\[ D^5 = \begin{bmatrix} 1^5 & 0 & 0 \\ 0 & 2^5 & 0 \\ 0 & 0 & (-1)^5 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 32 & 0 \\ 0 & 0 & -1 \end{bmatrix} \]
The final expression for \( A^5 \) needs further computation as:
\[ A^5 = PD^5P^{-1} \]
**Diagram/Matrix Layout**
To compute \( A^5 \), you would need to perform matrix multiplication using the given \( P \) and \( D^5 \) matrices and also compute the inverse of \( P \). The result, \( A^5 \), is depicted as a blank 3x3 matrix ready to be filled out after completing the necessary calculations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad35cd49-88f1-4759-9e4c-232c0c792d3e%2F7f8711e2-e741-4dc7-b52e-4ef4c212f8a4%2Ft4tjflid_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Matrix Exponentiation Using Diagonalization**
In this example, we are asked to compute \( A^5 \) given that the matrix \( A \) can be expressed in the form \( A = PDP^{-1} \).
Here, we have:
\[ P = \begin{bmatrix} 1 & 4 & 1 \\ 0 & -1 & 3 \\ 0 & 0 & -1 \end{bmatrix} \]
\[ D = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1 \end{bmatrix} \]
**Matrix \( A^5 \)**
The power of matrix \( A \), specifically \( A^5 \), can be computed by using the property of diagonalization:
\[ A^5 = (PDP^{-1})^5 = PD^5P^{-1} \]
Here, \( D^5 \) involves raising the diagonal matrix \( D \) to the power of 5. Since \( D \) is diagonal, its powers can be computed by raising each of the diagonal elements to the power of 5:
\[ D^5 = \begin{bmatrix} 1^5 & 0 & 0 \\ 0 & 2^5 & 0 \\ 0 & 0 & (-1)^5 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 32 & 0 \\ 0 & 0 & -1 \end{bmatrix} \]
The final expression for \( A^5 \) needs further computation as:
\[ A^5 = PD^5P^{-1} \]
**Diagram/Matrix Layout**
To compute \( A^5 \), you would need to perform matrix multiplication using the given \( P \) and \( D^5 \) matrices and also compute the inverse of \( P \). The result, \( A^5 \), is depicted as a blank 3x3 matrix ready to be filled out after completing the necessary calculations.
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