1. Holt Algebra 2 4-1 Matrices and Data 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. Objectives Use matrices to display mathematical and real-world data. Find sums, differences, and scalar products of matrices.

2. Holt Algebra 2 4-1 Matrices and Data 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.

3. Holt Algebra 2 4-1 Matrices and Data 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.

4. Holt Algebra 2 4-1 Matrices and Data 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 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.

5. Holt Algebra 2 4-1 Matrices and Data 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

6. Holt Algebra 2 4-1 Matrices and Data Use matrix M to answer the questions below. 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?

7. Holt Algebra 2 4-1 Matrices and Data You can add or subtract two matrices only if they have the same dimensions.

8. Holt Algebra 2 4-1 Matrices and Data Add or subtract, if possible. 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 =

9. Holt Algebra 2 4-1 Matrices and Data X – Z Subtract 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 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 =

10. Holt Algebra 2 4-1 Matrices and Data 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. 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.

11. Holt Algebra 2 4-1 Matrices and Data Add or subtract if possible. 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 B – AD – B

12. Holt Algebra 2 4-1 Matrices and Data 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. P = 3 –2 1 0 2 –1 Q= 4 7 2 5 1 –1 R = 1 4 –2 3 0 4 Evaluate 2PEvaluate -4R

13. Holt Algebra 2 4-1 Matrices and Data 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

14. Holt Algebra 2 4-1 Matrices and Data 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 0.8. WHY??? 150 87.5 125 70 200 112.5 0.8= 150 87.5 125 70 200 112.5 30 17.5 25 14 40 22.5 – 120 70 100 56 160 90

15. Holt Algebra 2 4-1 Matrices and Data 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.

16. Holt Algebra 2 4-1 Matrices and Data 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

17. Holt Algebra 2 4-1 Matrices and Data 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.

18. Holt Algebra 2 4-1 Matrices and Data 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

19. Holt Algebra 2 4-1 Matrices and Data

20. Holt Algebra 2 4-1 Matrices and Data 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