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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![We are at a city named Freezo, close to the North Pole, where the days are either "warm" or "cold" (with
temperature either above of below freezing). Assume that if we have a cold day today, then tomorrow will
be also be cold with probability p (and warm with probability 1 – p), and if today is warm then tomorrow
will be warm with probability q (and cold with probability 1 – q). Here p and q are some numbers in the
interval (0, 1).
Consider a matrix
1- g
A =
1- p
(8a) [4 marks] Prove that A= 1 is an eigenvalue of A (for any choice of p and q).
d.
(8b) [4 marks] Set p = 3/4 and q=1/2. Find the eigenvector
of A corresponding to X = 1, such that a + y =1.
(8c) [2 marks] It turns out that the numbers x and y that you found in (8b) represent the long-term
probability that any given day will be cold and warm, respectively. We are planning a 30-day event
"Snowtopia" in 2031 that requires at least 18 cold days during the event. Is Freezo a suitable place to
host this event? Explain why.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2771a61f-ae23-4f8b-aad4-bee2bb7c0abc%2F5ff114ab-6068-4752-a972-88c097d5e0de%2Fmdb5j5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:We are at a city named Freezo, close to the North Pole, where the days are either "warm" or "cold" (with
temperature either above of below freezing). Assume that if we have a cold day today, then tomorrow will
be also be cold with probability p (and warm with probability 1 – p), and if today is warm then tomorrow
will be warm with probability q (and cold with probability 1 – q). Here p and q are some numbers in the
interval (0, 1).
Consider a matrix
1- g
A =
1- p
(8a) [4 marks] Prove that A= 1 is an eigenvalue of A (for any choice of p and q).
d.
(8b) [4 marks] Set p = 3/4 and q=1/2. Find the eigenvector
of A corresponding to X = 1, such that a + y =1.
(8c) [2 marks] It turns out that the numbers x and y that you found in (8b) represent the long-term
probability that any given day will be cold and warm, respectively. We are planning a 30-day event
"Snowtopia" in 2031 that requires at least 18 cold days during the event. Is Freezo a suitable place to
host this event? Explain why.
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