Suppose that, for a certain metropolitan area and the surrounding counties, during each 5-year period an average of 20% of the metropolitan population M moves to the surrounding counties and the rest remains. Similarly, suppose that in the same period, an average of 30% of the surrounding counties' population S moves to the metropolitan area and the rest remains. This population dynamic can be represented as the following matrix

where row 1 shows how the metropolitan population M changed (80% remained in M and 20% moved to S), and row 2 shows how the population of the surrounding counties changed. Currently the population is evenly divided between the two areas, which can be represented by the row matrix
P = [% in M % in S] = [50% 50%] = [0.5 0.5]
(a) Form the product PD and interpret its entries.
(b) Predictions of this population distribution after 10 years and after 15 years could be found from what matrix products?
(c) Suppose that at some future time, the populations reach the point where 60% live in the metropolitan area and 40% live in the surrounding counties. Find the predicted distribution 5 years later?

80% 20% ]M 0.3 30% 70%]s 0.2 0.8 0.7. D=