2. Warm Up Distribute. 1. 3(2x + y + 3z) 2. –1(x – y + 2) State the property illustrated. 3. (a + b) + c = a + (b + c) 4. p + q = q + p 6x + 3y + 9z –x + y – 2 Associative Property of Addition Commutative Property of Addition

3. Use matrices to display mathematical and real-world data. Find sums, differences, and scalar products of matrices. Objectives

4. matrix dimensions entry address scalar Vocabulary

5. The table shows the top scores for girls in barrel racing at the 2004 National High School Rodeo finals. The data can be presented in a table or a spreadsheet as rows and columns of numbers. You can also use a matrix to show table data. A matrix is a rectangular array of numbers enclosed in brackets.

6. Matrix A has two rows and three columns. A matrix with m rows and n columns has dimensions m  n, read “m by n,” and is called an m  n matrix. A has dimensions 2  3. Each value in a matrix is called an entry of the matrix.

7. The address of an entry is its location in a matrix, expressed by using the lower case matrix letter with row and column number as subscripts. The score 16.206 is located in row 2 column 1, so a 21 is 16.206.

8. The prices for different sandwiches are presented at right. Example 1: Displaying Data in Matrix Form 6 in9 in Roast beef$3.95$5.95 Turkey$3.75$5.60 Tuna$3.50$5.25 A. Display the data in matrix form. P = 3.95 5.95 3.75 5.60 3.50 5.25 B. What are the dimensions of P? P has three rows and two columns, so it is a 3  2 matrix.

9. Example 1: Displaying Data in Matrix Form C. What is entry P 32 ? What does is represent? D. What is the address of the entry 5.95? The entry at P 32, in row 3 column 2, is 5.25. It is the price of a 9 in. tuna sandwich. The entry 5.95 is at P 12. The prices for different sandwiches are presented at right. 6 in9 in Roast beef$3.95$5.95 Turkey$3.75$5.60 Tuna$3.50$5.25

10. Use matrix M to answer the questions below. Check It Out! Example 1 a. What are the dimensions of M? 3  4 11 m 14 and m 23 b. What is the entry at m 32 ? c. The entry 0 appears at what two addresses?

11. You can add or subtract two matrices only if they have the same dimensions.

12. Add or subtract, if possible. Example 2A: Finding Matrix Sums and Differences W + Y Add each corresponding entry. W =, 3 –2 1 0 Y =, 1 4 –2 3 X =, 4 7 2 5 1 –1 Z = 2 –2 3 1 0 4 W + Y = 3 –2 1 0 + 1 4 –2 3 = 3 + 1 –2 + 4 1 + (–2) 0 + 3 4 2 –1 3 =

13. X – Z Subtract each corresponding entry. Example 2B: Finding Matrix Sums and Differences W =, 3 –2 1 0 Y =, 1 4 –2 3 X =, 4 7 2 5 1 –1 Z = 2 –2 3 1 0 4 Add or subtract, if possible. X – Z = 4 7 2 5 1 –1 2 –2 3 1 0 4 – 2 9 –1 4 1 –5 =

14. X + Y X is a 2  3 matrix, and Y is a 2  2 matrix. Because X and Y do not have the same dimensions, they cannot be added. Example 2C: Finding Matrix Sums and Differences W =, 3 –2 1 0 Y =, 1 4 –2 3 X =, 4 7 2 5 1 –1 Z = 2 –2 3 1 0 4 Add or subtract, if possible.

15. Add or subtract if possible. Check It Out! Example 2A B + D A =, 4 –2 –3 10 2 6 B =, 4 –1 –5 3 2 8 C =, 3 2 0 –9 –5 14 D = 0 1 –3 3 0 10 Add each corresponding entry. B + D = + 4 –1 –5 3 2 8 0 1 –3 3 0 10 4 + 0 –1 + 1 –5 + (–3) 3 + 3 2 + 0 8 + 10 = 4 0 –8 6 2 18

16. B – A Add or subtract if possible. Check It Out! Example 2B A =, 4 –2 –3 10 2 6 B =, 4 –1 –5 3 2 8 C =, 3 2 0 –9 –5 14 D = 0 1 –3 3 0 10 B is a 2  3 matrix, and A is a 3  2 matrix. Because B and A do not have the same dimensions, they cannot be subtracted.

17. D – BSubtract corresponding entries. Add or subtract if possible. Check It Out! Example 2C A =, 4 –2 –3 10 2 6 B =, 4 –1 –5 3 2 8 C =, 3 2 0 –9 –5 14 D = 0 1 –3 3 0 10 0 1 –3 3 0 10 4 –1 –5 3 2 8 – –4 2 2 0 –2 2 = D – B =

18. You can multiply a matrix by a number, called a scalar. To find the product of a scalar and a matrix, or the scalar product, multiply each entry by the scalar.

19. Example 3: Business Application Shirt Prices T-shirtSweatshirt Small$7.50$15.00 Medium$8.00$17.50 Large$9.00$20.00 X-Large$10.00$22.50 Use a scalar product to find the prices if a 10% discount is applied to the prices above. You can multiply by 0.1 and subtract from the original numbers. 7.5 15 8 17.5 9 20 10 22.5 7.5 15 8 17.5 9 20 10 22.5 – 0.1= 7.5 15 8 17.5 9 20 10 22.5 0.75 1.5 0.8 1.75 0.9 2 1 2.25 – 6.75 13.50 7.20 15.75 8.10 18.00 9.00 20.25

20. Example 3 Continued The discount prices are shown in the table. Discount Shirt Prices T-shirtSweatshirt Small$6.75$13.50 Medium$7.20$15.75 Large$8.10$18.00 X-large$9.00$20.25

21. Check It Out! Example 3 Ticket Service Prices DaysPlazaBalcony 1—2$150$87.50 3—8$125$70.00 9—10$200$112.50 Use a scalar product to find the prices if a 20% discount is applied to the ticket service prices. You can multiply by 0.2 and subtract from the original numbers. 150 87.5 125 70 200 112.5 – 0.2= 150 87.5 125 70 200 112.5 150 87.5 125 70 200 112.5 30 17.5 25 14 40 22.5 – 120 70 100 56 160 90

22. Check It Out! Example 3 Continued Discount Ticket Service Prices DaysPlazaBalcony 1—2$120$70 3—8$100$56 9—10$160$90

23. Example 4A: Simplifying Matrix Expressions Evaluate 3P — Q, if possible. P = 3 –2 1 0 2 –1 Q= 4 7 2 5 1 –1 R = 1 4 –2 3 0 4 P and Q do not have the same dimensions; they cannot be subtracted after the scalar products are found.

24. Example 4B: Simplifying Matrix Expressions Evaluate 3R — P, if possible. P = 3 –2 1 0 2 –1 Q= 4 7 2 5 1 –1 R = 1 4 –2 3 0 4 = 3 1 4 –2 3 0 4 – 3 –2 1 0 2 –1 = 3(1) 3(4) 3(–2) 3(3) 3(0) 3(4) – 3 –2 1 0 2 –1 = 3 12 –6 9 0 12 – 3 –2 1 0 2 –1 0 14 –7 9 –2 13

25. Check It Out! Example 4a Evaluate 3B + 2C, if possible. D = [6 –3 8] A = 4 –2 –3 10 C = 3 2 0 –9 B = 4 –1 –5 3 2 8 B and C do not have the same dimensions; they cannot be added after the scalar products are found.

26. Check It Out! Example 4b D = [6 –3 8] A = 4 –2 –3 10 C = 3 2 0 –9 B = 4 –1 –5 3 2 8 Evaluate 2A – 3C, if possible. 4 –2 –3 10 = 2– 3– 3 3 2 0 –9 2(4) 2(–2) 2(–3) 2(10) =+ –3(3) –3(2) –3(0) –3(–9) 8 –4 –6 20 =+ –9 –6 0 27 = –1 –10 –6 47

27. = [6 –3 8] + 0.5[6 –3 8] Check It Out! Example 4c D = [6 –3 8] A = 4 –2 –3 10 C = 3 2 0 –9 B = 4 –1 –5 3 2 8 Evaluate D + 0.5D, if possible. = [6 –3 8] + [0.5(6) 0.5(–3) 0.5(8)] = [6 –3 8] + [3 –1.5 4] = [9 –4.5 12]

29. Lesson Quiz 1. What are the dimensions of A? 2. What is entry D 12 ? Evaluate if possible. 3. 2A — C4. C + 2D5. 10(2B + D) –2 3  2 Not possible