Each week, of people who wear home-made masks switch to store-bought masks, while the other stick with their home-made masks. At the same time, of people who wear store-bought masks switch to home-made masks, while the other continue to buy masks from a store. The function that maps the number of people who wear home-made masks and the number of people who wear store-bought masks this week, to the number of people wearing home-made masks and the number of people wearing store-bought

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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7. Each week, of people who wear home-made masks switch to store-bought masks, while
the other stick with their home-made masks. At the same time, of people who
wear store-bought masks switch to home-made masks, while the other continue to buy
masks from a store. The function that maps the number of people who wear home-made
masks and the number of people who wear store-bought masks this week, to the number
of people wearing home-made masks and the number of people wearing store-bought
masks next week, has matrix representation
A
2|TW|N
ICOA
You are given that the characteristic polynomial of this matrix is
-
det(A – XI) = (1 - x) ( 52₂ − A).
a.) Find a matrix P and a diagonal matrix D for which A = PDP-¹.
b.) Find the limiting matrix lim A".
n→∞
c.) If these mask buying trends continue, what will the long-term distribution of home-
made mask wearers and store-bought mask wearers be?
Transcribed Image Text:7. Each week, of people who wear home-made masks switch to store-bought masks, while the other stick with their home-made masks. At the same time, of people who wear store-bought masks switch to home-made masks, while the other continue to buy masks from a store. The function that maps the number of people who wear home-made masks and the number of people who wear store-bought masks this week, to the number of people wearing home-made masks and the number of people wearing store-bought masks next week, has matrix representation A 2|TW|N ICOA You are given that the characteristic polynomial of this matrix is - det(A – XI) = (1 - x) ( 52₂ − A). a.) Find a matrix P and a diagonal matrix D for which A = PDP-¹. b.) Find the limiting matrix lim A". n→∞ c.) If these mask buying trends continue, what will the long-term distribution of home- made mask wearers and store-bought mask wearers be?
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