4 1 Matrices and Data Objectives Use matrices

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

4 -1 Matrices and Data The prices for different sandwiches are presented at right. A. Display the data in matrix form. 6 in 9 in Roast beef $3. 95 $5. 95 Turkey $3. 75 $5. 60 Tuna $3. 50 $5. 25 3. 95 5. 95 P = 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. Holt Algebra 2

4 -1 Matrices and Data The prices for different sandwiches are presented at right. 6 in 9 in Roast beef $3. 95 $5. 95 Turkey $3. 75 $5. 60 Tuna $3. 50 $5. 25 C. What is entry P 32? What does is represent? The entry at P 32, in row 3 column 2, is 5. 25. It is the price of a 9 in. tuna sandwich. D. What is the address of the entry 5. 95? The entry 5. 95 is at P 12. Holt Algebra 2

4 -1 Matrices and Data Use matrix M to answer the questions below. a. What are the dimensions of M? b. What is the entry at m 32? 3 4 11 c. The entry 0 appears at what two addresses? m 14 and m 23 Holt Algebra 2

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

4 -1 Matrices and Data Add or subtract, if possible. W= 3 – 2 1 0 , X= 4 7 2 5 1 – 1 , Y= 1 4 – 2 3 , Z= 2 – 2 3 1 0 4 W+Y Add each corresponding entry. W+Y= Holt Algebra 2 3 – 2 1 0 + 1 4 – 2 3 = 3+1 1 + (– 2) – 2 + 4 0+3 = 4 2 – 1 3

4 -1 Matrices and Data Add or subtract, if possible. W= 3 – 2 1 0 , X= 4 7 2 5 1 – 1 , Y= 1 4 – 2 3 , Z= X–Z Subtract each corresponding entry. X–Z= Holt Algebra 2 4 7 2 5 1 – 2 – 2 3 1 4 0 = 2 9 – 1 4 1 – 5 2 – 2 3 1 0 4

4 -1 Matrices and Data Add or subtract, if possible. W= 3 – 2 1 0 , X= 4 7 2 5 1 – 1 , Y= 1 4 – 2 3 , Z= 2 – 2 3 1 0 4 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. Holt Algebra 2

4 -1 Matrices and Data Add or subtract if possible. 4 – 2 A = – 3 10 , B = 2 6 B+D Holt Algebra 2 4 – 1 – 5 3 2 8 3 , C = 0 B–A 2 – 9 , D = – 5 14 0 1 – 3 3 0 D–B 10

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. 3 P= 1 2 – 2 0 – 1 Evaluate 2 P Holt Algebra 2 Q= 4 7 2 5 1 – 1 1 4 R = – 2 3 0 4 Evaluate -4 R

4 -1 Matrices and Data Use a scalar product to find the prices if a 10% discount is applied to the prices above. Shirt Prices T-shirt Sweatshirt Small $7. 50 $15. 00 Medium $8. 00 $17. 50 Large $9. 00 $20. 00 $10. 00 $22. 50 X-Large You can multiply by 0. 1 and subtract from the original numbers. 7. 5 15 8 17. 5 9 20 10 22. 5 Holt Algebra 2 7. 5 – 0. 1 8 15 17. 5 9 20 10 22. 5 = 6. 75 13. 50 7. 5 15 7. 20 15. 75 8 17. 5 – 8. 10 18. 00 9 20 9. 00 20. 25 10 22. 5 0. 75 1. 5 0. 8 1. 75 0. 9 2 1 2. 25

4 -1 Matrices and Data Check It Out! Example 3 Use a scalar product to find the prices if a 20% discount is applied to the ticket service prices. Ticket Service Prices Days Plaza Balcony 1— 2 $150 $87. 50 3— 8 $125 $70. 00 9— 10 $200 $112. 50 You can multiply by 0. 8. WHY? ? ? 0. 8 Holt Algebra 2 150 87. 5 125 70 200 112. 5 120 87. 5 70 150 = 125 100 70 56 160 112. 5 90 200 30 17. 5 – 25 14 40 22. 5

4 -1 Matrices and Data 3 P= 1 2 – 2 0 – 1 Q= 4 7 2 5 1 – 1 1 4 R = – 2 3 0 4 Evaluate 3 P — Q, if possible. P and Q do not have the same dimensions; they cannot be subtracted after the scalar products are found. Holt Algebra 2

4 -1 Matrices and Data 3 P= 1 2 – 2 0 Q= – 1 4 7 2 5 1 – 1 1 4 – 2 3 R= 0 4 Evaluate 3 R — P, if possible. 1 4 = 3 – 2 3 0 Holt Algebra 2 4 3 – 2 – 1 0 2 – 1 3(1) 3 0 12 14 3(4) 3 – 2 = 3(– 2) – – 1 – 6 – 79 9 3(3) 0 1 0 3(0) 0 – 21213 3(4) 2 – 1

4 -1 Matrices and Data A= 4 – 2 – 3 10 B= 4 – 1 – 5 3 2 8 C= 3 2 0 – 9 D = [6 – 3 8] Evaluate 3 B + 2 C, if possible. B and C do not have the same dimensions; they cannot be added after the scalar products are found. Holt Algebra 2

4 -1 Matrices and Data A= 4 – 2 – 3 10 B= 4 – 1 – 5 3 2 8 C= 3 2 0 – 9 D = [6 – 3 8] Evaluate 2 A – 3 C, if possible. =2 4 – 2 – 3 10 – 3 3 2 0 – 9 = = Holt Algebra 2 2(4) 2(– 2) 2(– 3) 2(10) 8 – 4 – 6 20 + + – 3(3) – 3(2) – 3(0) – 3(– 9) – 9 – 6 0 27 = – 10 – 6 47

4 -1 Matrices and Data Holt Algebra 2

4 -1 Matrices and Data Lesson Quiz 1. What are the dimensions of A? 3 2 2. What is entry D 12? – 2 Evaluate if possible. 3. 2 A — C 4. C + 2 D Not possible Holt Algebra 2 5. 10(2 B + D)