# Answer to Question #3237 in Matrix | Tensor Analysis for C Nelson

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?

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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