**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.

Name: **Matrix Exponentiation Using Diagonalization**In this example, we ar
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: **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 & -