Matrices Pre-Test and Workshop ...

Studying for Discrete Math

(5) A washing powder is sold in regular and large boxes. Market research shows that 40% of the buyers of the regular size switch to the large size for their next purchase, and 18% of the buyers of the large size switch to the regular size for their next purchase. The original market share was 70% for the regular size and 30% for the large size.

(a) What is the market share for each size in the next round of purchases?(Show the transition matrix used.) (3 marks)

(b) In the long term, what is the market share for each size of this washing powder? (2 marks)

As we all know by now, our beloved Manitoba Moose are a team that is prone to streaky play. In other words, when they are winning, it seems like they will never lose again. On the other hand, there are times when they are losing that it seems like they will never win another game! Being the statistical sport that hockey is, Joe Scoreboard has made the following observations concerning the Moose. That is, when the Moose win a game, there is a 70% chance that they will win their next game. However, when the Moose lose a game, they have only a 40% chance of winning their next game. Use this data to answer the following questions.

Manitoba Moose: Winning Streaks and Losing Streaks

Having played 10 games, could the Moose actually have a record represented by this probability ? Why or why not ?

5. Notice that there is very little difference between the results after 5 games and after 10 games. What do these percentages represent?

6. After how many games does the state matrix stabilize (become constant) to 4 decimal places? Use matrices to ilustrate this stable state.

7. The Moose play an 80 game regular season. Predict their final win - loss record assuming that they win their first game and also if they lose their first game. How do the records compare? In this example, how important is the initial state?

Wickers Furniture has two retail stores, A and B. Their inventory for the number of sofas, recliners, dressers, and dinettes is given in the table below.

(d) Write a 4 x 4 square matrix where the elements of the main diagonal are the prices, and all other elements are zeros. Label this matrix D. Find AD. What do the values in matrix AD represent?

(c) Calculate AB to determine the value of the inventory at each store. Label the new matrix C, and save it.

(b) Write a 4 x 1 column matrix that shows the prices, and name it B.

(a) Write a 2 x 4 matrix for the inventory and name it A.

Sofas cost $800 each, recliners $525 each, dressers $650 each, and dinettes $750 each.

sofas recliners dressers dinettesStore A 7 12 10 19Store B 15 6 8 9

HOMEWORK

A large grain cleaning plant has five centres of operation identified as A, B, C, D, and E. These sites are connected with conveyor belts to move grain as follows:

(c) Can grain be moved from any site to any other site using no more than two trips on the conveyor belt? If not, then indicate which sites cannot be connected in this way. Explain how you determine this answer.

(b) Write matrix B to represent the number of ways grain can be moved between sites via one other site.

(a) Create a square matrix A to represent the direct routes between sites. A '1' represents a direct route, and a '0' represents no direct route.

A can move grain to B and DB can move grain to A and CC can move grain to DD can move grain to EE can move grain to A and C

HOMEWORK