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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.

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

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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.

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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 x 2 matrix.

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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

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Use matrix M to answer the questions below. Whiteboards a. What are the dimensions of M?3 x 4 11 m 14 and m 23 b. What is the entry at m 32 ? c. The entry 0 appears at what two addresses?

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

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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 =

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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 =

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X + Y X is a 2 x 3 matrix, and Y is a 2 x 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.

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Add or subtract if possible. Whiteboards 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

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B – A Add or subtract if possible. Whiteboards 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 x 3 matrix, and A is a 3 x 2 matrix. Because B and A do not have the same dimensions, they cannot be subtracted.

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D – BSubtract corresponding entries. Add or subtract if possible. Whiteboards 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 =

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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.

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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

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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

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Whiteboards Ticket Service Prices DaysPlazaBalcony 12$150$87.50 38$125$70.00 910$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

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Discount Ticket Service Prices DaysPlazaBalcony 12$120$70 38$100$56 910$160$90

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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.

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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

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Whiteboards 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.

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Whiteboards 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

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= [6 –3 8] + 0.5[6 –3 8] Whiteboards 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]

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Chapter 4 Systems of Linear Equations; Matrices

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