Science:Math Exam Resources/Courses/MATH152/April 2015/Question B 5 (b)/Solution 1

Represent the proportion of people who are infected as column vectors like so:

${\begin{pmatrix}{\text{ proportion of people who are not infected }}\\{\text{ proportion of people who are infected }}\end{pmatrix}}$

Because ten percent of the people are infected this week , the vector is

${\begin{pmatrix}0.9\\0.1\end{pmatrix}}$

To see the percentage of people infected in two weeks, we calculate

$M^{2}{\begin{pmatrix}0.9\\0.1\end{pmatrix}}$

By part (a),

$M={\begin{pmatrix}0.7&0.1\\0.3&0.9\end{pmatrix}}$

So

$M^{2}{\begin{pmatrix}0.9\\0.1\end{pmatrix}}={\begin{pmatrix}0.7&0.1\\0.3&0.9\end{pmatrix}}^{2}{\begin{pmatrix}0.9\\0.1\end{pmatrix}}={\begin{pmatrix}0.7&0.1\\0.3&0.9\end{pmatrix}}{\begin{pmatrix}0.7&0.1\\0.3&0.9\end{pmatrix}}{\begin{pmatrix}0.9\\0.1\end{pmatrix}}={\begin{pmatrix}0.52&0.16\\0.48&0.84\end{pmatrix}}{\begin{pmatrix}0.9\\0.1\end{pmatrix}}={\begin{pmatrix}0.484\\0.516\end{pmatrix}}$

Hence, the proportion of the population that will be infected in two weeks is

$\color {blue}51.6{\text{ percent}}$