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c) Find the eigenvectors associated with each eigenvalue of M. d) Diagonalize M, i.e., write M in the form M = NDN-1, where N and D are 2 x 2 matrices and D is diagonal.

c) Find the eigenvectors associated with each eigenvalue of M. d) Diagonalize M, i.e., write M in the form M = NDN-1, where N and D are 2 x 2 matrices and D is diagonal.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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C, D And E please!

Each year, 5% of the population in Antapolis immigrate to Bumbleton, and 3% of the population in Bumbleton immigrate to Antapolis. Let ci, r; be the population of Antapolis and Bumbleton in the th year. In the following your mission is to find the distribution of the population. a) Find a 2 x 2 matrix M = such that for all natural numbers n ≥ 2 a b (d) ( ₂ ) = M (²n-1). Tn b) Find the eigenvalues of M. c) Find the eigenvectors associated with each eigenvalue of M. d) Diagonalize M, i.e., write M in the form M = NDN-1, where N and D are 2 x 2 matrices and D is diagonal. e) Let c₁ = r₁=0.5 (where we count population in units of millions). Derive a formula for Cn, Tn from (d), and find the equilibrium distribution, namely find limno Cn, limn→∞0 Tn.
Transcribed Image Text:Each year, 5% of the population in Antapolis immigrate to Bumbleton, and 3% of the population in Bumbleton immigrate to Antapolis. Let ci, r; be the population of Antapolis and Bumbleton in the th year. In the following your mission is to find the distribution of the population. a) Find a 2 x 2 matrix M = such that for all natural numbers n ≥ 2 a b (d) ( ₂ ) = M (²n-1). Tn b) Find the eigenvalues of M. c) Find the eigenvectors associated with each eigenvalue of M. d) Diagonalize M, i.e., write M in the form M = NDN-1, where N and D are 2 x 2 matrices and D is diagonal. e) Let c₁ = r₁=0.5 (where we count population in units of millions). Derive a formula for Cn, Tn from (d), and find the equilibrium distribution, namely find limno Cn, limn→∞0 Tn.
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