Answer
See explanation
Work Step by Step
The population movement problems in this section let the total population is constant, with no migration or immigration. The statement that "about $7 \%$ of the city's population moves to the suburbs" means that the rest of the city's population $(93 \%)$ remain in the city. This determines the entries in the first column of the migration matrix which concerns movement from the city.
\[
\begin{array}{l}
\text { From } \\
\text { City Suburbs To: } \\
{\left[\begin{array}{ccc}
0.93 & & \text {City} \\
0.07 & \text {Suburbs}
\end{array}\right.}
\end{array}
\]
Likewise, if $5 \%$ of the suburban population moves to the city, then the other $95 \%$ remain in the suburbs.
\[
M=\left[\begin{array}{ll}
0.93 & 0.05 \\
0.07 & 0.95
\end{array}\right]
\]
This determines the second column of the migration
matrix
The difference equation is
\[
x_{k+1}=M x_{k} \text { for } k=0,1,2, \ldots .
\]
Also,
\[
\begin{array}{r}
x_{0}=\left[\begin{array}{rr}
800,000 \\
500,000
\end{array}\right] \\
x_{1}=M x_{0}=\left[\begin{array}{rr}
0.93 & 0.05 \\
0.07 & 0.95
\end{array}\right]\left[\begin{array}{r}
800,000 \\
500,000
\end{array}\right]=\left[\begin{array}{r}
769,000 \\
531,000
\end{array}\right]
\end{array}
\]
The population in 2016 (when $k=1$ )
\[
x_{2}=M x_{1}=\left[\begin{array}{cc}
0.93 & 0.05 \\
0.07 & 0.95
\end{array}\right]\left[\begin{array}{c}
769,000 \\
531,000
\end{array}\right]=\left[\begin{array}{c}
741,172 \\
558,280
\end{array}\right]
\]
The population in 2002 (when $k=2$ )
