The three major shopping centres in a large city, Eastmall `(E)`, Grandmall `(G)` and Westmall `(W)`, are owned by the same company.
The total number of shoppers at each of the centres at 1.00 pm on a typical day is shown in matrix `V`.
`qquad qquad qquad {:(qquad qquad qquad \ E qquad qquad G qquad qquad \ W),(V = [(2300,2700,2200)]):}`
- Write down the order of matrix `V`. (1 mark)
Each of these centres has three major shopping areas: food `(F)`, clothing `(C)` and merchandise `(M)`.
The proportion of shoppers in each of these three areas at 1.00 pm on a typical day is the same at all three centres and is given in matrix `P` below.
`qquad qquad qquad P = [(0.48), (0.27), (0.25)] {:(F),(C),(M):}`
- Grandmall’s management would like to see 700 shoppers in its merchandise area at 1.00 pm.
If this were to happen, how many shoppers, in total, would be at Grandmall at this time? (1 mark)
- The matrix `Q = P xx V` is shown below. Two of the elements of this matrix are missing.
`{:(quad qquad qquad qquad \ E qquad qquad G qquad qquad W), (Q = [(1104, \ text{___}, 1056 ), (621,\ text{___}, 594), (575, 675, 550)]{:(F),(C), (M):}):}`
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- Complete matrix `Q` above by filling in the missing elements. (1 mark)
- The element in row `i` and column `j` of matrix `Q` is `q_(ij)`.
What does the element `q_23` represent? (1 mark)
The average daily amount spent, in dollars, by each shopper in each of the three areas at Grandmall in 2019 is shown in matrix `A_2019` below.
`qquad qquad A_2019 = [(21.30), (34.00), (14.70)] {:(F),(C),(M):}`
On one particular day, 135 shoppers spent the average daily amount on food, 143 shoppers spent the average daily amount on clothing and 131 shoppers spent the average daily amount on merchandise.
- Write a matrix calculation, using matrix `A_2019`, showing that the total amount spent by all these shoppers is $9663.20 (1 mark)
- In 2020, the average daily amount spent by each shopper was expected to change by the percentage shown in the table below.
Area food clothing merchandise Expected change increase by 5% decrease by 15% decrease by 1%
The average daily amount, in dollars, expected to be spent in each area in 2020 can be determined by forming the matrix product
`qquad qquad A_2020 = K xx A_2019`
Write down matrix `K`. (1 mark)