Science:Math Exam Resources/Courses/MATH152/April 2016/Question B 04 (d)

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MATH152 April 2016

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Question B 04 (d)
Suppose in the year 2020, 50 million people live in cities and 50 million in the suburbs. Every year, 10% of city residents move to the suburbs and 20% of the residents of the suburbs move to cities. (d) Assuming the overall population does not change (i.e., remains at 100 million), how many people will be living in the suburbs far in the future?

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Hint
We should compute $P^{t}$ for large $t.$ In other words, we need to find $\lim \limits _{t\to \infty }P^{t}.$ In order to do this, we consider "Matrix Diagonalization" to write $P=QAQ^{-1},$ and therefore $P^{t}=QA^{t}Q^{-1}.$

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Solution
We first use the Matrix diagonalization method to write the matrix $P$ in the following form: $P=QAQ^{-1},$ where $A={\begin{pmatrix}\lambda _{1}&0\\0&\lambda _{2}\end{pmatrix}}$ and $V={\begin{pmatrix}v_{1}{\Big \|}v_{2}\end{pmatrix}}.$ From part (a) and (c), we have $\lambda _{1}=0.7$ and $v_{1}={\begin{pmatrix}1\\-1\end{pmatrix}}.$ Also, $\lambda _{2}=1$ and $v_{2}={\begin{pmatrix}2\\1\end{pmatrix}}.$ So, we have $Q={\begin{pmatrix}1&2\\-1&1\end{pmatrix}},$ and therefore, $Q^{-1}={\frac {1}{3}}{\begin{pmatrix}1&-2\\1&1\end{pmatrix}}.$ We finally get that $P={\begin{pmatrix}0.9&0.2\\0.1&0.8\end{pmatrix}}=QAQ^{-1}={\begin{pmatrix}1&2\\-1&1\end{pmatrix}}{\begin{pmatrix}0.7&0\\0&1\end{pmatrix}}{\begin{pmatrix}{\frac {1}{3}}&-{\frac {2}{3}}\\{\frac {1}{3}}&{\frac {1}{3}}\end{pmatrix}},$ which means for any time $t$ we have $P^{t}={\begin{pmatrix}0.9&0.2\\0.1&0.8\end{pmatrix}}^{t}=QA^{t}Q^{-1}={\begin{pmatrix}1&2\\-1&1\end{pmatrix}}{\begin{pmatrix}0.7&0\\0&1\end{pmatrix}}^{t}{\begin{pmatrix}{\frac {1}{3}}&-{\frac {2}{3}}\\{\frac {1}{3}}&{\frac {1}{3}}\end{pmatrix}}={\begin{pmatrix}1&2\\-1&1\end{pmatrix}}{\begin{pmatrix}0.7^{t}&0\\0&1^{t}\end{pmatrix}}{\begin{pmatrix}{\frac {1}{3}}&-{\frac {2}{3}}\\{\frac {1}{3}}&{\frac {1}{3}}\end{pmatrix}}.$ Now we are ready to find $P^{t}$ for large $t$ by computing $\lim \limits _{t\to \infty }P^{t}.$ So, $\lim \limits _{t\to \infty }P^{t}=\lim \limits _{t\to \infty }QA^{t}Q^{-1}=\lim \limits _{t\to \infty }{\begin{pmatrix}1&2\\-1&1\end{pmatrix}}{\begin{pmatrix}0.7^{t}&0\\0&1^{t}\end{pmatrix}}{\begin{pmatrix}{\frac {1}{3}}&-{\frac {2}{3}}\\{\frac {1}{3}}&{\frac {1}{3}}\end{pmatrix}}={\begin{pmatrix}1&2\\-1&1\end{pmatrix}}{\begin{pmatrix}0&0\\0&1\end{pmatrix}}{\begin{pmatrix}{\frac {1}{3}}&-{\frac {2}{3}}\\{\frac {1}{3}}&{\frac {1}{3}}\end{pmatrix}}={\begin{pmatrix}1&2\\-1&1\end{pmatrix}}{\begin{pmatrix}0&0\\{\frac {1}{3}}&{\frac {1}{3}}\end{pmatrix}}={\frac {1}{3}}{\begin{pmatrix}2&2\\1&1\end{pmatrix}}.$ Now by applying this matrix on our initial population ${\begin{pmatrix}50\\50\end{pmatrix}},$ we get ${\frac {1}{3}}{\begin{pmatrix}2&2\\1&1\end{pmatrix}}{\begin{pmatrix}50\\50\end{pmatrix}}={\begin{pmatrix}{\frac {200}{3}}\\{\frac {100}{3}}\end{pmatrix}}.$ Therefore, $\color {blue}{\frac {100}{3}}$ million people will be living in the suburbs far in the future.