Transcribed Image Text:**Matrix Exponentiation using Diagonalization**
To compute \( A^k \), where \( k \) represents an arbitrary positive integer, we utilize the factorization \( A = PDP^{-1} \).
Given:
\[ A = \begin{bmatrix} 9 & -12 \\ 4 & -5 \end{bmatrix} \]
This matrix is expressed in the form \( A = PDP^{-1} \), where:
\[ P = \begin{bmatrix} 3 & 2 \\ 2 & 1 \end{bmatrix} \]
\[ D = \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix} \]
\[ P^{-1} = \begin{bmatrix} -1 & 2 \\ 2 & -3 \end{bmatrix} \]
To find \( A^k \), recognize the properties of diagonal matrices:
\[ A^k = (PDP^{-1})^k = PD^kP^{-1} \]
The computation simplifies significantly because raising a diagonal matrix to a power involves simply raising each diagonal entry to that power:
\[ D^k = \begin{bmatrix} 1^k & 0 \\ 0 & 3^k \end{bmatrix} \]
Finally, substitute back to compute \( A^k = P D^k P^{-1} \).
*Note*: The box or the placeholder indicates where the result of \( A^k \text{ would } \text{normally}\) be displayed once computed.
Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.
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![**Matrix Exponentiation using Diagonalization**
To compute \( A^k \), where \( k \) represents an arbitrary positive integer, we utilize the factorization \( A = PDP^{-1} \).
Given:
\[ A = \begin{bmatrix} 9 & -12 \\ 4 & -5 \end{bmatrix} \]
This matrix is expressed in the form \( A = PDP^{-1} \), where:
\[ P = \begin{bmatrix} 3 & 2 \\ 2 & 1 \end{bmatrix} \]
\[ D = \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix} \]
\[ P^{-1} = \begin{bmatrix} -1 & 2 \\ 2 & -3 \end{bmatrix} \]
To find \( A^k \), recognize the properties of diagonal matrices:
\[ A^k = (PDP^{-1})^k = PD^kP^{-1} \]
The computation simplifies significantly because raising a diagonal matrix to a power involves simply raising each diagonal entry to that power:
\[ D^k = \begin{bmatrix} 1^k & 0 \\ 0 & 3^k \end{bmatrix} \]
Finally, substitute back to compute \( A^k = P D^k P^{-1} \).
*Note*: The box or the placeholder indicates where the result of \( A^k \text{ would } \text{normally}\) be displayed once computed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee22cf2f-b974-4b00-a3cf-09b388e7d65d%2Ff7c4a280-d7b8-45e8-a4e0-34555f653113%2Fsu8hhaq_processed.jpeg&w=3840&q=75)
