Answer
See solution
Work Step by Step
\[
\begin{array}{l}
M=\left[\begin{array}{cc}
0.95 & 0.03 \\
0.05 & 0.97
\end{array}\right] \\
\mathbf{x}_{0}=\left[\begin{array}{c}
600,000 \\
400,000
\end{array}\right]
\end{array}
\]
The order of entries in a column of a migration matrix must match the order of the columns. For instance, if the first column concerns the population in the city, then the first entry in each column must be the fraction of the population that moves to the city.
\[
\begin{array}{l}
x_{5}=\left[\begin{array}{c}
523,293 \\
476,707
\end{array}\right] \\
x_{10}=\left[\begin{array}{c}
472,737 \\
527,263
\end{array}\right] \\
x_{15}=\left[\begin{array}{c}
439,417 \\
560,583 \\
417,456 \\
582,544
\end{array}\right] \\
x_{20}=\left[\begin{array}{c}
4 \\
50
\end{array}\right]
\end{array}
\]
Some of the population vectors are given.
The data here shows that the city population is declining and the suburban population is increasing, but the changes in population each year grow smaller.
\[
\begin{array}{c}
\mathrm{x}_{0}=\left[\begin{array}{c}
350,000 \\
650,000
\end{array}\right] \\
x_{5}=\left[\begin{array}{c}
358,523 \\
641,477
\end{array}\right] \\
x_{10}=\left[\begin{array}{c}
364,140 \\
635,860
\end{array}\right] \\
x_{15}=\left[\begin{array}{c}
367,843 \\
632,157 \\
370,283 \\
629,717
\end{array}\right] \\
x_{20}=
\end{array}
\]
When
\[
\mathbf{x}_{0}=\left[\begin{array}{l}
350,000 \\
650,000
\end{array}\right]
\]
the situation is different. The city population is increasing slowly and the suburban population is decreasing. No other conclusions are expected
