A coach purchases 47 hats for his players and their families at a total cost of $302. The cost of a small hat is $5.50. A medium hat costs $6.00. A large hat costs $7.00. He purchases three times as many medium hats as small hats. Using matrices, how many large hats did the coach purchase?

a. 6

b. 11

c. 18

d. 23

First, we define variables:

x: small hat

y: medium hat

z: large hat

We now write the system of equations:

x + y + z = 47

5.50x + 6y + 7z = 302

-3x + y = 0

We can write the system in matrix form as:

Ax = b

Where,

A = [1 1 1; 5.50 6 7; - 3 1 0]

b = [47; 302; 0]

x = [x; y; z]

Solving the system we have:

x = 6

y = 18

z = 23

Answer:

the coach did purchase 23 large hats

d. 23