Four families decided to go watch a movie on a Friday evening. The Rahmans (R) have 6 family members, the Khans (K) have 4, Sarkars (S) 2, and the Hasans (H) 2.

They buy a movie-ticket for each member. During the intermission, R goes to the concession-stand and buys 4 large popcorns. K buys 3. S and the H buy 2 and 0 respectively. Once the movie ends, the families go to an ice-cream parlour nearby. R orders 5 scoops of ice-cream and 2 coffees. K orders 2 scoops of ice-cream and 1 coffees. S orders 3 scoops of ice-cream and 1 coffees. H orders 2 scoops of ice-creams and 2 coffees.

At the end of the night, the total expenditure of the families are as follows: The Rahmans have spent Tk. 5775; the Khans have spent Tk. 3489; the Sarkars have spent Tk. 2506; and the Hasans have spent Tk. 2074.

Using the information above, convert the Friday night expenditures of the four families into a system of linear equations. Convert this system of equations into the matrix form.

What is the determinant of the coefficient matrix?

What is the price of a movie ticket?

What is the price of a pop-corn?

What is the price of a scoop of ice-cream?

What is the price of a cup of coffee?

Suppose next week (week 2), the four families go to the movies again. But this time, two things are different: i. the concession-stand is not open and the families cannot buy the pop-corns and ii. the price of the other goods have changed from what it was last week. If I give you the information about which family bought how much of each good, and what their total expenditure is, will it be possible to calculate the price of a movie-ticket, a scoop to ice-cream, and a cup of coffee?

Suppose next week (week 3), only three of the families go to the movies. This time, the concession-stand is open and the three families once again buy movie-tickets, pop-corns, ice-creams, and coffees. However, the price of goods have changed again. If I gave you the information about which family buys how much of each good, and what their total expenditure is, will it be possible to calculate the price of the four goods?