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?

Let s, m, and l represent the numbers of small, medium, and large hats. The equations are s + m + l = 47, m = 3s, and 5.50s + 6m + 7l = 302; these can be written as a matrix system and solved. Substituting m = 3s and l = 47 − 4s into the cost equation gives 329 − 4.50s = 302, so s = 6. Thus l = 47 − 4(6) = 23, so the coach bought 23 large hats.

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