smc-618-60-2×2 Matrix

MATRICES, FUR1 2021 VCAA 2 MC

Every Friday, the same number of workers from a large office building regularly purchase their lunch from one of two locations: the deli, `D `, or the cafe, `C`.

It has been found that:

- of the workers who purchase lunch from the deli on one Friday, 65% will return to purchase from the deli on the next Friday
- of the workers who purchase lunch from the cafe on one Friday, 55% will return to purchase from the cafe on the next Friday.

A transition matrix that can be used to describe this situation is

A. `qquad text(this Friday)` `{:(qquad\ D quadquad \ C quad),([(0.55,0.35),(0.45,0.65)]{:(D),(C):} qquad text(next Friday)):}`	B. `qquad text(this Friday)` `{:(qquad\ D quadquad \ C quad),([(0.65,0.45),(0.45,0.55)]{:(D),(C):} qquad text(next Friday)):}`

C. `qquad text(this Friday)` `{:(qquad\ D quadquad \ C quad),([(0.65,0.55),(0.45,0.55)]{:(D),(C):} qquad text(next Friday)):}`	D. `qquad text(this Friday)` `{:(qquad\ D quadquad \ C quad),([(0.65,0.45),(0.35,0.55)]{:(D),(C):} qquad text(next Friday)):}`

E. `qquad text(this Friday)` `{:(qquad\ D quadquad \ C quad),([(0.65,0.55),(0.35,0.45)]{:(D),(C):} qquad text(next Friday)):}`

Show Answers Only

`D`

Show Worked Solution

`text{65% of workers return to deli}`

`=> e_11 = 0.65`

`text{55% of workers return to cafe}`

`=> e_22 = 0.55`

`text{Column elements must sum to 1}`

`=> D`

MATRICES, FUR2-NHT 2019 VCAA 3

The basketball finals will be televised on `C_3` from 12.00 noon until 4.00 pm.

It is expected that 600 Gillen residents will be watching `C_3` at any time from 12.00 noon until 4.00 pm.

The remaining 1400 Gillen residents will not be watching `C_3` from 12.00 noon until 4.00 pm (represented by NotC₃).

`{:(qquadqquadqquadquadtext(this hour)),(qquadqquadqquad \ C_3 quadquad \ NotC_3),(P = [(v, qquad quad w quad),(0.35, quad qquad x quad)]{:(C_3),(NotC_3):}\ text(next hour)):}`

Write down the values of `v, w` and `x` in the boxes provided below. (2 marks)

`v =`

`w =`

`x =`

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Show Answers Only

`v = 0.65, \ w = 0.15, \ x = 0.85`

Show Worked Solution

`v = 1-0.35 = 0.65`

`text(Residents who change from)\ C_3\ text(to Not)C_3`

`= 0.35 xx 600`

`= 210`

`text(S) text(ince 600 residents are watching)\ C_3\ text(at any time)`

`w xx 1400`	`= 210`
`w`	`= 0.15`
`:. x`	`= 1-0.15`
	`= 0.85`

MATRICES, FUR1-NHT 2019 VCAA 5 MC

A population of birds feeds at two different locations, `A` and `B`, on an island.

The change in the percentage of the birds at each location from year to year can be determined from the transition matrix `T` shown below.

`{:(),(),(T=):}{:(qquadtext(this year)),((qquadA,\ B)),([(0.8,0.4),(0.2,0.6)]):}{:(),(),({:(A),(B):}):}{:(),(),(text(next year)):}`

In 2018, 55% of the birds fed at location `B`.

In 2019, the percentage of the birds that are expected to feed at location `A` is

Show Answers Only

`D`

Show Worked Solution

`[(A_2019),(B_2019)] = [(0.8,0.4),(0.2,0.6)][(0.45),(0.55)] = [(0.58),(0.42)]`

`=>\ D`

MATRICES, FUR2 2010 VCAA 4

The Dinosaurs (`D`) and the Scorpions (`S`) are two basketball teams that play in different leagues in the same city.

The matrix `A_1` is the attendance matrix for the first game. This matrix shows the number of people who attended the first Dinosaur game and the number of people who attended the first Scorpion game.

`A_1 = [(2000),(1000)]{:(D),(S):}`

The number of people expected to attend the second game for each team can be determined using the matrix equation

`A_2 = GA_1`

where `G` is the matrix `{:(qquadqquadqquadtext(this game)),((qquadqquadqquadD,qquad\ S)),(G = [(1.2,-0.3),(0.2,0.7)]{:(D),(S):}qquad{:text(next game):}):}`

1. Determine `A_2`, the attendance matrix for the second game. (1 mark)
  
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2. Every person who attends either the second Dinosaur game or the second Scorpion game will be given a free cap. How many caps, in total, are expected to be given away? (1 mark)
  
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Assume that the attendance matrices for successive games can be determined as follows.

`A_3 = GA_2`

`A_4 = GA_3`

and so on such that `A_(n + 1) = GA_n`

Determine the attendance matrix (with the elements written correct to the nearest whole number) for game 10. (1 mark)

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Describe the way in which the number of people attending the Dinosaurs’ games is expected to change over the next 80 or so games. (1 mark)

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The attendance at the first Dinosaur game was 2000 people and the attendance at the first Scorpion game was 1000 people.

Suppose, instead, that 2000 people attend the first Dinosaur game, and 1800 people attend the first Scorpion game.

Describe the way in which the number of people attending the Dinosaurs’ games is expected to change over the next 80 or so games. (1 mark)

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Show Answers Only

1. `A_2 = [(2100),(1100)]`
2. `3200\ text(people)`
`A_10 = [(2613),(1613)]`
`text(Attendance at the Dinosaur’s games increases gradually)`

`text(to 3000, at which level it remains steady.)`
`text(Attendence at the Dinosaur’s games decreases)`

`text(gradually to 600, where it remains steady.)`

Show Worked Solution

a.i.	`A_2`	`= GA_1`
		`= [(1.2,-0.3),(0.2,0.7)][(2000),(1000)]`
		`= [(2100),(1100)]`

a.ii. `text(Total attending second games)`

`= 2100 + 1100`

`= 3200\ text(people)`

b.	`A_10`	`= GA_9`
		`= G^9A_1`
		`= [(1.2,-0.3),(0.2,0.7)]^9[(2000),(1000)]`
		`= [(2613),(1613)]`

`A_80`

`= G^79A_1= [(3000),(2000)]`

`A_81`

`= G^80A_1= [(3000),(2000)]`

`:.\ text{Attendance at the Dinosaur’s games increases gradually to 3000,}`

`text{at which level it remains steady.}`

d. `text(Using the new initial attendences,)`

`A_80 = [(1.2,-0.3),(0.2,0.7)]^79[(2000),(1800)] = [(600),(400)]`

`A_81 = [(1.2,-0.3),(0.2,0.7)]^80[(2000),(1800)] = [(600),(400)]`

`:.\ text{Attendence at the Dinosaur’s games decreases gradually to 600,}`

`text{where it remains steady.}`

MATRICES, FUR1 2009 VCAA 9 MC

`T =[[0.8,0.3],[0.2,0.7]]` is a transition matrix.

`S_3 = [[1150],[850]]` is a state matrix.

If `S_3 = TS_2`, then `S_2` equals

A. `[[1000],[1000]]`

B. `[[1090],[940]]`

C. `[[1100],[900]]`

D. `[[1150],[850]]`

E. `[[1175],[825]]`

Show Answers Only

`C`

Show Worked Solution

`S_3`	`= TS_2`
`:. S_2`	`= T^(−1)S_3`
	`= [(0.8,0.3),(0.2,0.7)]^(−1)[(1150),(850)]`
	`= [(1.4,−0.6),(−0.4,1.6)][(1150),(850)]`
	`= [(1100),(900)]`

`=> C`

MATRICES, FUR1 2009 VCAA 7-8 MC

In a country town, people only have the choice of doing their food shopping at a store called Marks (`M`) or at a newly opened store called Foodies (`F`).

In the first week that Foodies opened, only 300 of the town’s 800 shoppers did their food shopping at Marks. The remainder did their food shopping at Foodies.

Part 1

A state matrix `S_1` that can be used to represent this situation is

A. `S_1 = [[300],[800]]{:(M),(F):}`

B. `S_1 = [[500],[300]]{:(M),(F):}`

C. `S_1 = [[800],[300]]{:(M),(F):}`

D. `S_1 = [[300],[500]]{:(M),(F):}`

E. `S_1 = [[800],[500]]{:(M),(F):}`

Part 2

A market researcher predicts that

- of those who do their food shopping at Marks this week, 70% will shop at Marks next week and 30% will shop at Foodies
- of those who do their food shopping at Foodies this week, 90% will shop at Foodies next week and 10% will shop at Marks.

A transition matrix that can be used to represent this situation is

Show Answers Only

`text(Part 1:)\ D`

`text(Part 2:)\ B`

Show Worked Solution

`text(Part 1)`

`=> D`

`text(Part 2)`

`text(Columns must add up to 1.0,)`

`:.\ text(Eliminate)\ C\ text(and)\ D.`

`text(The information that 90% of Foodies)`

`text(shoppers stay means that)\ \ e_(FF) = 0.90.`

`:.\ text(Eliminate)\ A\ text(and)\ E.`

`=> B`

MATRICES, FUR1 2009 VCAA 6 MC

`T` is a transition matrix, where

`{:(qquadqquadqquadquadtext(from)),({:qquadqquadqquad\ PqquadQ:}),(T = [(0.6,0.7),(0.4,0.3)]{:(P),(Q):}{:qquadtext(to):}):}`

An equivalent transition diagram, with proportions expressed as percentages, is

Show Answers Only

`C`

Show Worked Solution

`text(The loop at)\ P\ text(is 60% and)\ Q\ text(is 30%.)`

`=> C`

MATRICES, FUR1 2010 VCAA 7 MC

A new colony of several hundred birds is established on a remote island. The birds can feed at two locations, A and B. The birds are expected to change feeding locations each day according to the transition matrix

`{:(qquad qquad qquad {:text(this day):}), (qquad qquad qquad {: A\ \ \ \ \ B:}), (T = [(0.4, 0.3),(0.6, 0.7)] {:(A), (B):} {:quad text(next day):}):}`

In the beginning, approximately equal numbers of birds feed at each site each day.

Which of the following statements is not true

A. 70% of the birds that feed at B on a given day will feed at B the next day

B. 60% of the birds that feed at A on a given day will feed at B the next day.

C. In the long term, more birds will feed at B than at A.

D. The number of birds that change feeding locations each day will decrease over time to zero

E. In the long term, some birds will always be found feeding at each location.

Show Answers Only

`D`

Show Worked Solution

`A, B, and C\ \ text(are all true.)`

`text(Consider)\ \ D and E,`

`text(If)\ \ n\ \ text{is large (say}\ n = 100 text{),}`

`[(0.4, 0.3), (0.6, 0.7)]^100 = [(0.333, 0.333), (0.666, 0.666)]`

`:.\ text(Birds will always be feeding at both locations and)`

`text(some will be changing locations each day.)`

`=> D`

MATRICES, FUR1 2011 VCAA 5-6 MC

Two politicians, Rob and Anna, are the only candidates for a forthcoming election. At the beginning of the election campaign, people were asked for whom they planned to vote. The numbers were as follows.

During the election campaign, it is expected that people may change the candidate that they plan to vote for each week according to the following transition diagram.

Part 1

The total number of people who are expected to change the candidate that they plan to vote for one week after the election campaign begins is

A. `828`

B. `1423`

C. `2251`

D. `4269`

E. `6891`

Part 2

The election campaign will run for ten weeks.

If people continue to follow this pattern of changing the candidate they plan to vote for, the expected winner after ten weeks will be

A. Rob by about 50 votes.

B. Rob by about 100 votes.

C. Rob by fewer than 10 votes.

D. Anna by about 100 votes.

E. Anna by about 200 votes.

Show Answers Only

`text(Part 1:)\ C`

`text(Part 2:)\ E`

Show Worked Solution

`text(Part 1)`

`text(Students expected to change)`

`= 25text(%) xx 5692 + 24text(%) xx 3450`

`= 2251`

`=> C`

`text(Part 2)`

♦ Mean mark 48%.

`text(After 1 week,)`

`[(0.75,0.24),(0.25,0.76)][(5692),(3450)] = [(5097),(4045)]`

`text(After 10 weeks,)`

`[(0.75,0.24),(0.25,0.76)]^10[(5692),(3450)] = [(4479),(4663)]`

`:.\ text(Anna is ahead by)`

`4663 – 4479 = 184\ text(votes)`

`=> E`

MATRICES, FUR1 2012 VCAA 5 MC

There are two fast-food shops in a country town: Big Burgers (B) and Fast Fries (F).

Every week, each family in the town will purchase takeaway food from one of these shops.

The transition diagram below shows the way families in the town change their preferences for fast food from one week to the next.

A transition matrix that provides the same information as the transition diagram is

Show Answers Only

`D`

Show Worked Solution

`{:(qquadqquadqquadqquadquad\ text(from)),({:qquadqquadqquadqquad\ BqquadquadF:}),( :. T = [(0.8,0.3),(0.2,0.7)]{:(B),(F):}qquadtext(to)):}`

`rArr D`

`text(Tuesday)`	`= [(0.85, 0.35),(0.15, 0.65)][(0.40),(0.60)]`
`text(Friday)`	`= [(0.85, 0.35),(0.15, 0.65)]^4 [(0.40),(0.60)]`
	`~~ [(0.68),(0.32)]`