More on Matrices

1. More on Matrices Mary Dwyer Wolfe, Ph.D. Macon State College MSP with Bibb County Schools November 2010

2. Performance Standard • MM3A7. Students will understand and apply matrix representations of vertex-edge graphs. a. Use graphs to represent realistic situations. b. Use matrices to represent graphs, and solve problems that can be represented by graphs.

3. Learning Task • An Okefenokee Food Web Learning Task

4. Okeefenokee Web Learning Task Construct the associated matrix F to represent this web. What does a row containing a single one indicate? One source of food. What does a column of zeros indicate? No predators.

5. Okeefenokee Web Learning Task 2. Which animals have the most direct sources of food? How can this be determined from the matrix? Find the number of direct food sources for each animal.

6. Okeefenokee Web Learning Task 3. The insect column has the most ones. What does this suggest about the food web?

7. Okeefenokee Web Learning Task 4. The matrix F2 denotes indirect (through one intermediary) sources of food. For example, the fish relies on insects for food, and the bear relies on the fish for food, so the insect is an indirect source of food for the bear. Find F2. Notice that insect column contains all nonzero numbers. What does this indicate? Matrix F Matrix F2

8. Okeefenokee Web Learning Task 5. Compute additional powers of the food web matrix to represent the number of direct and indirect sources of food for each animal. Which animal has the most food sources? Matrix F Matrix F3

9. Okeefenokee Web Learning Task If an insecticide is introduced into the food web, killing the entire insect population, several animals will lose a source of food. 6. Construct a new matrix G to represent the food web with no insects. What effect does this have on the overall animal population? What has happened to the row sums? Compare these with those of the original matrix. What does a row sum of zero indicate? Matrix F Matrix G

10. Okeefenokee Web Learning Task 7. Will all the animals be affected by the insecticide? Which animal(s) will be least affected? Matrix F Matrix G

11. Okeefenokee Web Learning Task 7. Will all the animals be affected by the insecticide? Which animal(s) will be least affected? Matrix G Matrix G2

12. Okeefenokee Web Learning Task 8. Organize and summarize your findings in a brief report to the health officials. Take and support a position on whether using an insecticide to destroy the insect population is harmful to the environment.

13. Solving Systems of Equations • MM3A4. Students will perform basic operations with matrices. • a. Add, subtract, multiply, and invert matrices, when possible, choosing appropriate methods, including technology. • b. Find the inverses of two-by-two matrices using pencil and paper, and find inverses of larger matrices using technology. • c. Examine the properties of matrices, contrasting them with properties of real numbers. • MM3A5. Students will use matrices to formulate and solve problems. • a. Represent a system of linear equations as a matrix equation. • b. Solve matrix equations using inverse matrices. • c. Represent and solve realistic problems using systems of linear equations.

14. Solving Systems of Equations • A movie theater sells tickets for $9.00 each. Senior citizens receive a discount of $3.00. One evening the theater sold 636 tickets and took in $4974 in revenue. How many tickets were sold to senior citizens? How many were sold to “moviegoers” who were not senior citizens? Let x = number of senior citizen tickets sold Let y = numbers of non-senior citizen tickets sold x + y = 636 Total tickets sold 6x + 9y = 4974 Total money both kinds of tickets.

15. Solving Systems of Equations • A movie theater sells tickets for $9.00 each. Senior citizens receive a discount of $3.00. One evening the theater sold 636 tickets and took in $4974 in revenue. How many tickets were sold to senior citizens? How many were sold to “moviegoers” who were not senior citizens? Let x = number of senior citizen tickets sold Let y = numbers of non-senior citizen tickets sold 386 tickets were sold to moviegoers who were not senior citizens.

16. Solving Systems of Equations • A movie theater sells tickets for $9.00 each. Senior citizens receive a discount of $3.00. One evening the theater sold 636 tickets and took in $4974 in revenue. How many tickets were sold to senior citizens? How many were sold to “moviegoers” who were not senior citizens? Let x = number of senior citizen tickets sold Let y = numbers of non-senior citizen tickets sold 386 tickets were sold to moviegoers who were not senior citizens.

17. Solving Systems of Equations • John inherited $25,000 and invested part of it in a money market account, part in municipal bonds, and part in a mutual fund. After one year, he received a total of $1,620 in simple interest from the three investments. The money market paid 6% annually, the bonds paid 7% annually, and the mutually fund paid 8% annually. There was $6,000 more invested in the bonds than the mutual funds. Find the amount John invested in each category. Let x = amount invested in a money market Let y = amount invested in municipal bonds Let z = amount invested in a mutual fund x + y + z = 25,000 0.06x + 0.07y + 0.08z = 1620 y = z + 6000

18. Solving Systems of Equations x + y + z = 25,000 0.06x + 0.07y + 0.08z = 1620 y = z + 6000 Rewrite as: x + y + z = 25,000 0.06x + 0.07y + 0.08z = 1620 y – z = 6000

19. Solving Systems of Equations

20. Solving Systems of Equations Let x = amount invested in a money market Let y = amount invested in municipal bonds Let z = amount invested in a mutual fund $15,000 was invested in a money market, $8000 in municipal bonds and $2000 in a mutual fund.