The car park at a theme park has three areas, `A, B` and `C`.
The number of empty `(E)` and full `(F)` parking spaces in each of the three areas at 1 pm on Friday are shown in matrix `Q` below.
`{:(qquad qquad qquad \ E qquad F),(Q = [(70, 50),(30, 20),(40, 40)]{:(A),(B),(C):}quad text(area)):}`
- What is the order of matrix `Q`? (1 mark)
- Write down a calculation to show that 110 parking spaces are full at 1 pm. (1 mark)
Drivers must pay a parking fee for each hour of parking.
Matrix `P`, below, shows the hourly fee, in dollars, for a car parked in each of the three areas.
`{:(qquad qquad qquad qquad qquad text{area}), (qquad qquad qquad A qquad quad quad B qquad qquad C), (Q = [(1.30, 3.50, 1.80)]):}`
- The total parking fee, in dollars, collected from these 110 parked cars if they were parked for one hour is calculated as follows.
`qquad qquad qquad P xx L = [207.00]`
where matrix `L` is a `3 xx 1` matrix.
Write down matrix `L`. (1 mark)
The number of whole hours that each of the 110 cars had been parked was recorded at 1 pm. Matrix `R`, below, shows the number of cars parked for one, two, three or four hours in each of the areas `A, B` and `C`.
`{:(qquadqquadqquadqquadquadtext(area)),(quad qquadqquadquad \ A qquad B qquad C),(R = [(3, 1, 1),(6, 10, 3),(22, 7,10),(19, 2, 26)]{:(1),(2),(3),(4):}\ text(hours)):}`
- Matrix `R^T` is the transpose of matrix `R`.
Complete the matrix `R^T` below. (1 mark)
`qquad R^T = [( , , , , , , , , ), ( , , , , , , , , ), ( , , , , , , , , ), ( , , , , , , , , ), ( , , , , , , , , )]`
- Explain what the element in row 3, column 2 of matrix `R^T` represents. (1 mark)