### Matrix Powers and Calculations**Consider the following matrix:**\[P = \begin{bmatrix} 1 & 0 \\ 0.8 & 0.2 \end{bmatrix}\]---**(a) Calculate $ P^2 = P \cdot P $.**- **Matrix Structure**: Two blank boxes are presented for entries of matrix $ P^2 $.- **Incorrect Mark**: A red "X" indicates that the calculation was incorrect.---**(b) Calculate $ P^4 = P^2 \cdot P^2 $.**- **Matrix Structure**: Two blank boxes are presented for entries of matrix $ P^4 $.- **Incorrect Mark**: A red "X" signifies another incorrect calculation.---**(c) Calculate $ P^8 $. (Round all entries to four decimal places.)**- **Matrix Result**: The correct entries have been filled in as: \[ \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \]- **Correct Mark**: The absence of a red "X" here suggests the calculation was performed correctly.---**(d) Without computing it explicitly, find $ P^{1,000} $. (Round all entries to four decimal places.)**- **Matrix Result**: The expected entries mimic the result of $ P^8 $: \[ \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \]---### Explanation of DiagramsThe task explores the power of a matrix and involves multiplying matrices to certain powers, specifically $ P^2 $, $ P^4 $, $ P^8 $, and extrapolating to $ P^{1,000} $. Each matrix operation visually introduces boxed matrices where entries are calculated. Feedback is given through red “X” marks for incorrect answers, while the absence of such indicates correctness. The calculations emphasize accuracy and rounding to four decimal places for certain powers.

Answered: Image /qna-images/answer/c4805a93-0895-40c0-ab71-2f255ea81df6.jpg

Answered: Consider the following matrix. 1 P =…

Consider the following matrix. 1 P = 0.8 0.2 (a) Calculate P2 = P. P. (b) Calculate Pp4 = p2 . p2. (c) Calculate Pº. (Round all entries to four decimal places.) 1

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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Question

### Matrix Powers and Calculations

**Consider the following matrix:**

\[
P = \begin{bmatrix} 1 & 0 \\ 0.8 & 0.2 \end{bmatrix}
\]

---

**(a) Calculate $ P^2 = P \cdot P $.**

- **Matrix Structure**: Two blank boxes are presented for entries of matrix $ P^2 $.
- **Incorrect Mark**: A red "X" indicates that the calculation was incorrect.

---

**(b) Calculate $ P^4 = P^2 \cdot P^2 $.**

- **Matrix Structure**: Two blank boxes are presented for entries of matrix $ P^4 $.
- **Incorrect Mark**: A red "X" signifies another incorrect calculation.

---

**(c) Calculate $ P^8 $. (Round all entries to four decimal places.)**

- **Matrix Result**: The correct entries have been filled in as:
\[
\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}
\]

- **Correct Mark**: The absence of a red "X" here suggests the calculation was performed correctly.

---

**(d) Without computing it explicitly, find $ P^{1,000} $. (Round all entries to four decimal places.)**

- **Matrix Result**: The expected entries mimic the result of $ P^8 $:
\[
\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}
\]

---

### Explanation of Diagrams

The task explores the power of a matrix and involves multiplying matrices to certain powers, specifically $ P^2 $, $ P^4 $, $ P^8 $, and extrapolating to $ P^{1,000} $. Each matrix operation visually introduces boxed matrices where entries are calculated. Feedback is given through red “X” marks for incorrect answers, while the absence of such indicates correctness. The calculations emphasize accuracy and rounding to four decimal places for certain powers.

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