Science:Math Exam Resources/Courses/MATH152/April 2016/Question B 04 (b)/Solution 1

The matrix $P={\begin{pmatrix}0.9&0.2\\0.1&0.8\end{pmatrix}}$ shows the transition of each year. The initial population of city and suburb can be written as ${\begin{pmatrix}50&50\end{pmatrix}}^{T}.$ Since we need to find the population of city in 2022 ( two years after 2020), we compute the following:

${\begin{pmatrix}0.9&0.2\\0.1&0.8\end{pmatrix}}{\begin{pmatrix}0.9&0.2\\0.1&0.8\end{pmatrix}}{\begin{pmatrix}50\\50\end{pmatrix}}={\begin{pmatrix}0.9&0.2\\0.1&0.8\end{pmatrix}}{\begin{pmatrix}55\\45\end{pmatrix}}={\begin{pmatrix}58.5\\41.5\end{pmatrix}}.$

So, the population of city is 58.5 million in 2022. So, the fraction is ${\frac {58.5}{100}}=0.585.$