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

Vocabulary matrix dimensions entry address scalar

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.

Matrix A has two rows and three columns 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.

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 a21 is 16.206.

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

Example 1: Displaying Data in Matrix Form Roast beef $3.95 $5.95 Turkey $3.75 $5.60 Tuna $3.50 $5.25 The prices for different sandwiches are presented at right. C. What is entry P32? What does is represent? The entry at P32, 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 P12.

Use matrix M to answer the questions below. Check It Out! Example 1 Use matrix M to answer the questions below. a. What are the dimensions of M? 3  4 b. What is the entry at m32? 11 c. The entry 0 appears at what two addresses? m14 and m23

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

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

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

Example 2C: Finding Matrix Sums and Differences Add or subtract, if possible. 3 –2 1 0 4 7 2 5 1 –1 1 4 –2 3 2 –2 3 1 0 4 W = , X = , Y = , Z = 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.

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

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

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

You can multiply a matrix by a number, called a scalar 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.

Example 3: Business Application 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 X-Large $10.00 $22.50 You can multiply by 0.1 and subtract from the original numbers. 6.75 13.50 7.20 15.75 8.10 18.00 9.00 20.25 7.5 15 8 17.5 9 20 10 22.5 7.5 15 8 17.5 9 20 10 22.5 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 – – 0.1 =

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

You can multiply by 0.2 and subtract from the original numbers. Check It Out! Example 3 Ticket Service Prices Days Plaza Balcony 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 70 200 112.5 150 87.5 125 70 200 112.5 150 87.5 125 70 200 112.5 120 70 100 56 160 90 30 17.5 25 14 40 22.5 – – 0.2 =

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

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

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

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

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

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