algebra 2 matrices problem

Question

Each swimmer is allowed to compete in 7 events at a swim meet.

1. Danny got first place 3 times, second place 1 time, third twice, and fourth once.
2. John got first 2 times, second place 3 times, third twice, and didn’t place fourth.
3. Kenny didn’t place first, but was second once, third four times, and fourth two times.
4. Sean placed first once, second twice, third three times, and fourth once.

The club teams are waiting to see which trophy they will be getting. Their trophy is based on the amount of points each team received.

1. The Eagles Aquatics team had 8 people get first place, 9 in second, 10 in third, and 5 in fourth.
2. The Pirates swim team had 6 people get first place, 7 in second, 3 in third, and 1 in fourth.
3. Devil Rays Aquatic Club had 9 people get first, 2 in second, 5 in third, and 7 in fourth.
4. The Manta Rays swim team had 4 first-place winners, 3 second, 2 third, and 1 fourth.
5. Create a matrix for the individual swimmers and name it A. What are the dimensions?
6. Create a matrix for the teams and name it B. What are the dimensions?
7. Create a matrix for the points awarded and name it C. What are the dimensions?
8. First place receives 6 points, second 4, third 3, and fourth 2.
9. Which swimmer got the most points?
10. Which team won the trophy?
11. Explain how you got your answers.
12. Does the order in which you multiply the matrices affect your results? Explain.
13. For the first two meets of the season, Danny, John, Kenny, and Sean had the exact same results. Use scalar multiplication to determine the totals of first, second, third, and fourth place for each individual swimmer.
14. Look back at what you did for questions 4 and 5. How does this relate to your understanding of vectors?
15. The top three teams qualified to compete at the state competition. Use the following information to create a system of equations where x is the points for first place, y is the points for second place, and z is the points for third place.
16. Results:
17. Eagles got 5 first-place, 3 second-place, and 4 third-place finishes with a team total of 67 points.
18. Pirates got 2 first-place, 5 second-place, and 5 third-place finishes with a team total of 56.
19. Devil Rays got 6 first-place, 3 second-place and 3 third-place finishes with a team total of 72.

We can’t believe the Devil Rays came from behind and won! We need to determine how many points were awarded for each place. Use a method of your choice (Cramer’s rule or the inverse method) to get the answers. Explain why you chose that particular method.

Answer 1

This algebraic problem demonstrates how matrices simplify the application of Algebra to sets of variables. You could simply tally the points of each players and team without relying on matrices. But, the Algebra of matrices tackle problems with huge data sizes, such as voice and optical recognitions problems....

Here, we are only dealing with 4 players and 4 teams, each awarded on point scale of 4 steps. In contrast, voice and optical recognition involve thousands of such crossly related variables that need to be factored, reversed, and ordered by the Algebra of matrices.

Our problem is summarized in the following 3 lists or tables:

A. Players Matrix 4 x 4

1st place	2nd place	3rd place	4th place
Danny	3	1	2	1
John	2	3	2	0
Kenny	0	1	4	2
Sean	1	2	3	1

B. Team Matrix 4 x 4

1st place	2nd place	3rd place	4th place
Eagles Aquatics	8	9	10	5
Pirates	6	7	3	1
Devil Rays Aquatic	9	2	5	7
Manta Rays	4	3	2	1

C. Points Matrix 4 x 1

Points
1st place	6
2nd place	4
3rd place	3
4th place	2

Whether you opt to rely on simple Algebra by {multiplying frequency of places by points of each place }

or, you opt to use the rules of multiplication of Matrices, you will sure accomplish the same outcome.

For Players: (4 x 4) players' frequency of winning per place x (4 x 1) points matrix = 4 x 1 gained points

For Teams: (4 x 4) team' s frequency of winning per place x (4 x 1) points matrix = 4 x 1 gained points

In order to realize that matrix multiplication is not commutative (i.e., 4 x4 matrix A and 4 x 4 matrix B could only be multiplied by 4x1 matrix C as A x C = 4 x 1, or B x C = 4 x 1). That is justified by the logics of scaler multiplication given the following tables:

A1. Players’ Points Matrix 4 x 1

1st place	2nd place	3rd place	4th place
Danny	3 x 6	1 x 4	2 x 3	1 x 2	= 30
John	2 x 6	3 x 4	2 x 3	0 x 2	= 30
Kenny	0 x 6	1 x 4	4 x 3	2 x 2	= 20
Sean	1 x 6	2 x 4	3 x 3	1 x 2	= 25
=36	=28	= 33	= 8

B1. Team’s Points Matrix 4 x 1

1st place	2nd place	3rd place	4th place
Eagles Aquatics	8 x 6	9 x 4	10 x 3	5 x 2	= 124
Pirates	6 x 6	7 x 4	3 x 3	1 x 2	= 75
Devil Rays Aquatic	9 x 6	2 x 4	5 x 3	7 x 2	= 91
Manta Rays	4 x 6	3 x 4	2 x 3	1 x 2	= 44
= 162	= 84	= 60	= 28

We could not perform the multiplication C x A , nor C x B, because C is 4 x1, and both A and B are 4 x 4 ranks.

Results:

Points	Rank
Danny	= 30	1
John	= 30	1
Kenny	= 20	2
Sean	= 25	3
Points	Rank
Eagles Aquatics	= 124	1
Pirates	= 75	2
Devil Rays Aquatic	= 91	3
Manta Rays	= 44	4