## Example 10

### Basketball Performance Factors (Matrix Operations)

Students should be able to use matrices to help them solve a wide variety of mathematics problems. The following elementary problem may be used to introduce students to the use of matrices to organize data and to the use of matrix operations for solving problems.

Problem. The women's basketball team at the University of Connecticut completed a perfect 35-0 season by defeating the University of Tennessee to win the 1995 NCAA women's basketball championship. Senior Rebecca Lobo led her team in scoring with 17 points. Does the fact that she contributed 17 points indicate the total worth of her contribution to the victory? Does the fact that Nykesha Sales scored only 10 points indicate that she made much less of a contribution? Obviously, other performance factors contribute to a team's effort. Positive factors include assists (a), steals (s), rebounds (r), and blocked shots (b). On the other hand, turnovers (t) and personal fouls (f) are examples of negative factors. Here is a listing of some of the nonshooting performance factors for the seven Connecticut players who participated in the championship game.

 a s r b t f Elliott Lobo Wolters Rizzotti Webber Sales Berube 3 2 0 3 2 3 2 1 0 0 3 0 3 0 7 8 3 3 1 6 3 0 2 2 0 0 0 0 5 2 1 4 0 1 3 3 4 4 3 1 3 0

a. Let each instance of a positive factor count 1 point and each instance of a negative factor count -1 point. Find each player's nonshooting performance score.

b. Now think of the box score as a 7 x 6 matrix A and similarly the incentive clause as a 6 x 1 matrix B:

Use your calculator or computer to find the product AB. How does the 7 x 1 product matrix compare with the calculations made in part a?

c. Based on your observations here, explain how matrix multiplication works. Try your ideas out by multiplying the following on your calculator and comparing it to your hand or mental calculations. Make a statement about the relative dimensions of the factor matrices and the size of the resulting product.

i.

ii.

iii.

Explain any "strangeness" you observe here.

Generalizations and Extensions

d. The owner of the New York Yankees baseball team has developed a uniform incentive clause for hitters on his team. He has decided to pay them \$100 for each time at bat, \$400 for each run scored, \$300 for each hit, and \$500 for each run batted in. Find the sports section in a newspaper and calculate the value of the incentive clause for one game for the first six hitters in the Yankees' lineup. If it is not baseball season, go to the library and find the results for an old game. Set up two matrices, the 6x4 performance matrix and the 4x1 incentive-clause matrix, so that your calculator can do the multiplication for you.

e. In a series of games, you can calculate the value of the incentive clause for each game and then add them together. As an alternative, explain how the calculator could be used to add the performance statistics matrices of all of the games before multiplying the incentive matrix. In other words, explain how matrix addition should work.

f. Compare and contrast matrix addition with real number addition. Also, compare and contrast matrix multiplication with real number multiplication. Include a discussion of various properties such as associativity, commutativity, existence of identities and inverses, and the distributive property.