Question #3237

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?

Expert's answer

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

.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

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