Question

The transition matrix shows the percentage of car owners who will switch for their next purchase:

NA JAP GER OTH
NA .55 .20 .15 .10
JAP .20 .60 .10 .10
GER .35 .10 .45 .10
OTH .20 .30 .10 .40

If the present ownership in Toronto is:
NA - 50 000
JAP - 20 000
GER - 17000
OTH - 12000
What will the members be:

a) next year b) In two years?

Answer 1

Let A =
.55 .20 .15 .10
.20 .60 .10 .10
.35 .10 .45 .10
.20 .30 .10 .40

be the transition matrix and

B =
50000
20000
17000
12000

be the vector of present ownership.
Then the vector of members in next year can be calculated as a product of matrix A
to the vector B:

A * B =
35250
24900
28350
22500

and the vector of members in two years is

A*A*B =
30870
27075
29835
26355