1. Price-Quantity Example In January, the Left Coast Bookstore chain sold 700 hardcover books, 1,400 softcover books, and 2,500 plastic books in San Francisco; it sold 300 hardcover, 500 softcover, and 500 plastic books in Los Angeles. Now, hardcover books sell for $30 each, softcover books sell for $10 each, and plastic books sell for $15 each. Suppose that each hardcover book costs the stores $10, each softcover book costs $5, and each plastic book costs $10. Use matrix operations to compute the total profit at each store in January.

2. Silent Appointment #1 You have to prepare two bouquets out of roses, carnations and lilies. The cost of a rose is $4, cost of a carnation is $2.5 and that of a lily is $5. If your friend is willing to split the cost of the second bouquet evenly with you, use matrix multiplication to determine how much you'll have to spend. RosesCarnationsLilies First Bouquet675 Second Bouquet 864

3. Problem Solving Strategies/Steps Read the Entire Problem Thoroughly Identify what you are ultimately looking for in the problem. Identify and underline the relevant information you will/may use in solving the problem Determine what the relevant information means Set the problem up using the relevant information and relationships Solve the problem Check to ensure the solution actually “finds” what you were looking for in the problem.

4. JIGSAW ACTIVITY by ROW In 1966, Washington (Redskins) and New York (Giants) played the highest scoring game in National Football League history. The table below summarizes the scoring. A touchdown (TD) is worth 6 points, a field goal (FG) is worth three points, a safety (S) is worth 2 points, and point after touchdown (PAT) is worth one point. Using matrix multiplication, what was the final score? TDFGSPAT Redskins 10109 Giants 6005

5. Problem Solving 2x 3=3x + 23 -3 -7x+y -3 -4x Find x and y using the matrices above

6. Investment – Return Example Ally invested $10,000 in a mutual fund guaranteed to return 8.2% annual interest. She also invested $5,000 in a high risk oil fund that could earn up to 15% interest annually and then balanced that out by investing in a government savings bond worth $20,000 but only returns 2% annually. Assuming everything turns out as planned on her investments, how much money will she have at the end of the year?

7. SILENT APPOINTMENT ACTIVITY Jamar invested a total of $100,000 in four investments. The first earns 5% interest annually so he invested half in that safe mutual fund. He invested half of what he had left in government bonds returning 3.5% and the remaining he split evenly between a savings and a checking account returning 1.5% and 1% annually respectively. At the end of a year, how much money did he have?

8. 9 3x+1=916 2y-1 10-510 Find x and y in the matrices above

9. JIGSAW ACTIVITY A nut distributor wants to know the nutritional content of various mixtures of almonds, cashews, and pecans. Her supplier has provided the following nutrition information: Her first mixture, a protein blend, consists of 6 cups of almonds, 3 cups of cashews, and 1 cup of pecans. Her second mixture, a low fat mix, consists of 3 cups of almonds, 6 cups of cashews, and 1 cup of pecans. Her third mixture, a low carb mix consists of 3 cups of almonds, 1 cup of cashews, and 6 cups of pecans. Determine the amount of protein, carbs, and fats in a 1 cup serving of each of the mixtures. AlmondsCashewsPecans Protein g/cup26.22110.1 Carbs g/cup40.244.814.3 Fat g/cup71.963.582.8

10. Silent Appointment #3 Sometimes you’ll get a matrix word problem where just numbers are given; these are pretty tricky. Here is one: An outbreak of Chicken Pox hit the local public schools. Approximately 15% of the male and female juniors and 25% of the male and female seniors are currently healthy, 35% of the male and female juniors and 30% of the male and female seniors are currently sick, and 50% of the male and female juniors and 45% of the male and female seniors are carriers of Chicken Pox. There are 100 male juniors, 80 male seniors, 120 female juniors, and 100 female seniors. Using two matrices and one matrix equation, find out how many males and how many females (don’t need to divide by class) are healthy, sick, and carriers.

11. Solution The best way to approach these types of problems is to set up a few manual calculations and see what we’re doing. For example, to find out how many healthy males we would have, we’d set up the following equation and do the calculation:.15(100) +.25(80) = 35. Likewise, to find out how many females are carriers, we can calculate:.50(120) +.45(100) = 105. We can tell that this looks like matrix multiplication. And since we want to end up with a matrix that has males and females by healthy, sick and carriers, we know it will be either a 2 x 3 or a 3 x 2. But since we know that we have both juniors and seniors with males and females, the first matrix will probably be a 2 x 2. That means, in order to do matrix multiplication, the second matrix that holds the %’s of students will have to be a 2 x 3, since there are 3 types of students, healthy (H), sick (S), and carriers (C). Notice how the percentages in the rows in the second matrix add up to 100%. Also notice that if we add up the number of students in the first matrix and the last matrix, we come up with 400.

12. Solution Continued jr.sr.HSC Male10080x Jr..15.35.50 Fem.120100 Sr..25.30.45 HSC = Jr.355986 Sr.4372105 So there will be 35 healthy males, 59 sick males, and 86 carrier males, 43 healthy females, 72 sick females, and 95 carrier females.

13. Warm-Up Let matrix:M =111 Calculate M 2, M 3, and M 4. Do you notice a pattern?

14. Identity Matrix 4 7a b 4 7 x = -5 2c d -5 2 So what do you think the identity matrix means? What do you think the identity matrix for a two by two matrix?

15. Identity Matrix When dealing with matrix computation, it is important to understand the identity matrix. Think of the identity matrix as the multiplicative identity of square matrices, or the one of square matrices. Any square matrix multiplied by the identity matrix of equal dimensions on the left or the right doesn't change.

16. Identity Matrix 4 71 0 4 7 x = -5 20 1 -5 2

17. Determinants Determinants are useful properties of square matrices, but can involve a lot of computation. A 2×2 determinant is much easier to compute than the determinants of larger matrices, like 3×3 matrices. To find a 2×2 determinant we use a simple formula that uses the entries of the 2×2 matrix. 2×2 determinants can be used to find the area of a parallelogram and to determine invertibility of a 2×2 matrix.

18. Determinants a b So if A is c d, then the determinant of A is given by the formulaI A I = ad – bc 2 1 So if B is, then the determinant of B…. IBI = 2x5 -1x3 3 5

19. Determinants So, if the Determinant of matrix A is 5 and A = 1 4 then, what is “x”? -2 x Remember IMI is equal to the difference of the diagonals in a two by two matrix….. 1x – (-8) = 5 so x + 8 = 5 and X = -3

20. Guided Practice Find the Determinants of each of the following matrices…… 7 9 0 -2 8 -7 -3 x 8 5 -4 7 4 3 -2 8

21. Inverse of a Matrix The inverse of a matrix is often used to solve matrix equations. It is also used in Matrix Division….

22. Finding the Inverse of a 2 x 2 Matrix Step 1 : Find the determinant. Step 2 : Swap the elements of the leading diagonal. Step 3: Change the signs of the elements of the other diagonal. Step 4: Divide each element by the determinant.

23. Finding the Inverse of a 2 x 2 Matrix So let’s find the inverse of this matrix…. P 7 8 P = 5 6 First the inverse of matrix P is written as P Step 1? Find the Determinant

24. Finding the Inverse of a 2 x 2 Matrix Step 2 : Swap the elements of the leading diagonal. so 7 8becomes 6 8 5 6 5 7

25. Finding the Inverse of a 2 x 2 Matrix Step 3: Change the signs of the elements of the other diagonal. So 7 8 becomes 6 -8 5 6 -5 7

26. Finding the Inverse of a 2 x 2 Matrix Step 4: Divide each element by the determinant. so the inverse of P, or P to the negative one Is 3-4 -2.53.5

27. Practice in Pairs Find the Inverse of this Matrix… 12-9 -8-4 7-13 x y