Chapter 3.4, Problem 52E

Demographics Suppose that a government study showed that $90\%$ of urban families remained in an urban area in the next generation (and $10\%$ moved to a rural area), whereas $70\%$ of rural families remained in a rural area in the next generation (and $30\%$ moved to an urban area). This means that if $\left[\begin{array}{c}{u}_0\\ r_0\end{array}\right]$ represents the current percents of urban families, $u_0$, and of rural families, $r_0$, then

$\left[\begin{array}{c}u\\ r\end{array}\right]=\left[\begin{array}{cc}0.90& 0.30\\ 0.10& 0.70\end{array}\right]\left[\begin{array}{c}{u}_0\\ r_0\end{array}\right]$

Represents the percents of urban families, $u$, and rural families, $r$, one generation from now. Currently the population is $60\%$ urban and $40\%$ rural. Find the percents in each group one generation before this one.