Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![In the image, we have a transition matrix \( P \) defined as:
\[
P = \begin{bmatrix} 1 & 0 \\ 0.5 & 0.5 \end{bmatrix}
\]
This matrix is employed to calculate its powers, illustrating how a system changes over successive states.
1. **Calculation of \( P^2 \):**
\[
P^2 = \begin{bmatrix} 1 & 0 \\ 0.75 & 0.25 \end{bmatrix}
\]
This square matrix results from multiplying the matrix \( P \) by itself. Each element of the resulting matrix indicates how the initial states transition considering two iterations.
2. **Calculation of \( P^4 \):**
\[
P^4 = \begin{bmatrix} 1 & 0 \\ 2.5 & 0.625 \end{bmatrix}
\]
For \( P^4 \), the matrix \( P \) is raised to the fourth power, signifying the state transitions across four steps or iterations. Note the changes in the lower half of the matrix, indicating cumulative transitions.
3. **Placeholder for \( P^5 \):**
\[
P^5 = \begin{bmatrix} \, & \, \\ \, & \, \end{bmatrix}
\]
This is a blank matrix suggesting that the calculation for \( P^5 \), the fifth power of \( P \), has yet to be completed or requires input.
These matrices are crucial for understanding state transitions in processes modeled by such matrices, often used in areas like Markov chains and dynamic systems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F603fbc9e-cdba-4fa0-b3dd-1f2ec72dd924%2F0bbcf1f0-1c82-4a02-abf7-c9533b0a22f6%2Fpk2n8g_processed.png&w=3840&q=75)
Transcribed Image Text:In the image, we have a transition matrix \( P \) defined as:
\[
P = \begin{bmatrix} 1 & 0 \\ 0.5 & 0.5 \end{bmatrix}
\]
This matrix is employed to calculate its powers, illustrating how a system changes over successive states.
1. **Calculation of \( P^2 \):**
\[
P^2 = \begin{bmatrix} 1 & 0 \\ 0.75 & 0.25 \end{bmatrix}
\]
This square matrix results from multiplying the matrix \( P \) by itself. Each element of the resulting matrix indicates how the initial states transition considering two iterations.
2. **Calculation of \( P^4 \):**
\[
P^4 = \begin{bmatrix} 1 & 0 \\ 2.5 & 0.625 \end{bmatrix}
\]
For \( P^4 \), the matrix \( P \) is raised to the fourth power, signifying the state transitions across four steps or iterations. Note the changes in the lower half of the matrix, indicating cumulative transitions.
3. **Placeholder for \( P^5 \):**
\[
P^5 = \begin{bmatrix} \, & \, \\ \, & \, \end{bmatrix}
\]
This is a blank matrix suggesting that the calculation for \( P^5 \), the fifth power of \( P \), has yet to be completed or requires input.
These matrices are crucial for understanding state transitions in processes modeled by such matrices, often used in areas like Markov chains and dynamic systems.
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