Perform Basic Matrix Operations

Transcription

1 TEKS 3.5 a.1, a. Perform Basic Matrix Operations Before You performed operations with real numbers. Now You will perform operations with matrices. Why? So you can organize sports data, as in Ex. 34. Key Vocabulary matrix dimensions elements equal matrices scalar scalar multiplication A matrix is a rectangular arrangement of numbers in rows and columns. For example, matrix A below has two rows and three columns. The dimensions of a matrix with m rows and n columns are m 3 n (read m by n ). So, the dimensions of matrix A are 3 3. The numbers in a matrix are its elements. A 5F G rows The element in the first row and third column is 5. 3 columns Two matrices are equal if their dimensions are the same and the elements in corresponding positions are equal. KEY CONCEPT For Your Notebook Adding and Subtracting Matrices To add or subtract two matrices, simply add or subtract elements in corresponding positions. You can add or subtract matrices only if they have the same dimensions. Adding Matrices F a b c dg 1F e f g hg 5F a 1 e b1 f c 1 g d1 hg Subtracting Matrices F a b c dg F e f g hg 5F a e b f c g d hg E XAMPLE 1 Add and subtract matrices AVOID ERRORS Be sure to verify that the dimensions of two matrices are equal before adding or subtracting them. Perform the indicated operation, if possible. a. F G 1F 1 4 0G 5F 3 1 (1) G 5F 4 b. F 7 6G 4 0 F G 3 1G 5F 1G 5F 7 () (10) 1 (3) G 3.5 Perform Basic Matrix Operations 187

2 SCALAR MULTIPLICATION In matrix algebra, a real number is often called a scalar. To multiply a matrix by a scalar, you multiply each element in the matrix by the scalar. This process is called scalar multiplication. E XAMPLE Multiply a matrix by a scalar COMPARE ORDER OF OPERATIONS The order of operations for matrix expressions is similar to that for real numbers. In particular, you perform scalar multiplication before matrix addition and subtraction. Perform the indicated operation, if possible. (1) 5F(4) (1) (0) 5F8 0 () (7)G 4 14G a. F G 8 b. 4F 5 0G 1F 3 8 4() 4(8) 6 5G5F 4(5) 4(0)G 1F G 8 3 5F 0 0G 1F G 8 1 (3) F (5)G F 6 5G GUIDED PRACTICE for Examples 1 and Perform the indicated operation, if possible F 4 6 8G 1F. 8 4G F GF G F G 3F G 1F 0 6G MATRIX PROPERTIES Many of the properties you have used with real numbers can be applied to matrices as well. CONCEPT SUMMARY For Your Notebook Properties of Matrix Operations Let A, B, and C be matrices with the same dimensions, and let k be a scalar. Associative Property of Addition (A 1 B) 1 C 5 A 1 (B 1 C) Commutative Property of Addition Distributive Property of Addition Distributive Property of Subtraction A 1 B 5 B 1 A k(a 1 B) 5 ka 1 kb k(a B) 5 ka kb 188 Chapter 3 Linear Systems and Matrices

3 ORGANIZING DATA Matrices are useful for organizing data and for performing the same operations on large numbers of data values. E XAMPLE 3 TAKS REASONING: Multi-Step Problem MANUFACTURING A company manufactures small and large steel DVD racks with wooden bases. Each size of rack is available in three types of wood: walnut, pine, and cherry. Sales of the racks for last month and this month are shown below. Small Rack Sales Walnut Pine Cherry Large Rack Sales Walnut Pine Cherry Last month This month Last month This month Organize the data using two matrices, one for last month s sales and one for this month s sales. Then write and interpret a matrix giving the average monthly sales for the two month period. Solution STEP 1 Organize the data using two 3 3 matrices, as shown. Walnut Pine Cherry Last Month (A) This Month (B) Small Large Small Large F G F G ANOTHER WAY You can also evaluate } 1 (A 1 B) by first using the distributive property to rewrite the expression as } 1 A 1 } 1 B. STEP STEP 3 Write a matrix for the average monthly sales by first adding A and B to find the total sales and then multipling the result by } }( A 1 B) 5 } SF G1F05 300GD 5 1 } F G 5F G Interpret the matrix from Step. The company sold an average of 110 small walnut racks, 107 large walnut racks, 97 small pine racks, 33 large pine racks, 15 small cherry racks, and 85 large cherry racks. 3.5 Perform Basic Matrix Operations 189

4 SOLVING MATRIX EQUATIONS You can use what you know about matrix operations and matrix equality to solve an equation involving matrices. E XAMPLE 4 Solve a matrix equation 5x 3SF 6 4G 1F 3 7 Solve the matrix equation for x and y. Solution Simplify the left side of the equation. 5x 3SF 6 4G 1F ygd 5F ygd 5F 3 4G Write original equation. 3 4G 3F 5x yg 5F Add matrices inside parentheses. 3 4G F15x yG 5F Perform scalar multiplication. 3 4G Equate corresponding elements and solve the two resulting equations. 15x y 5 4 x 5 y 5 4 c The solution is x 5 and y 5 4. GUIDED PRACTICE for Examples 3 and 4 5. In Example 3, find B A and explain what information this matrix gives. 3x 1 6. Solve SF 4 yg 1F GD 5F 18G for x and y. 3.5 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 5, 1, and 33 5 TAKS PRACTICE AND REASONING Exs. 8, 9, 33, 34, 36, and VOCABULARY Copy and complete: The? of a matrix with 3 rows and 4 columns are WRITING Descri be how to determine whether two matrices are equal. EXAMPLE 1 on p. 187 for Exs ERROR ANALYSIS Describe and correct the error in adding the matrices. F5G 1F 3.8G 5F G 190 Chapter 3 Linear Systems and Matrices

5 ADDING AND SUBTRACTING MATRICES Perform the indicated operation, if possible. If not possible, state the reason. 4. F G 1F G F 5 3G F 7. F G1F G G F G1F G 3 4G 6. F G F F GF 5G 6 EXAMPLE on p. 188 for Exs MULTIPLYING BY A SCALAR Perform the indicated operation. 10. F G 11. 3F G 1. 4F 3 5 } F G } F G MATRIX OPERATIONS Use matrices A, B, C, and D to evaluate the matrix expression B 3 1G 5F C 6 0G 5F D G 5F A 5F A 1 B 17. B A 18. 4A B 19. } 7 }4G.F G G } 3 B 0. C 1 D 1. C 1 3D. D C C D EXAMPLE 4 on p. 190 for Exs. 4 7 SOLVING MATRIX EQUATIONS Solve the matrix equation for x and y. 1 3x F 4 5G 5F y 5G 5. F x 6 1 8G 1 F G 5F F 8 x 5 6G F yG 5F G 7. 4xF 1 3 6G 5F yG 13 yg 8. TAKS REASONING Based on the equation below, what is the value of the expression 3x y? F x G 5F yG A 7.15 B 9.1 C 10.1 D Find two matrices A and B such that A 3B 5F 5 0 TAKS REASONING 30. CHALLENGE Find the matrix X that makes the equation true. a. X 1F G 5F G b. X F 3 5 0G 5F 8 6 c. X 1F G 5F d. 3X 0 10G F 1G 5F 1 G. 1 3G G 3.5 Perform Basic Matrix Operations 191

6 PROBLEM SOLVING EXAMPLE 3 on p. 189 for Exs SNOWBOARD SALES A sporting goods store sells snowboards in several different styles and lengths. The matrices below show the number of each type of snowboard sold in 003 and 004. Write a matrix giving the change in sales for each type of snowboard from 003 to 004. Freeride Alpine Freestyle Sales for 003 Sales for cm 155 cm 160 cm 165 cm 150 cm 155 cm 160 cm 165 cm F G F G 3. FUEL ECONOMY A car dealership sells four different models of cars. The fuel economy (in miles per gallon) is shown below for each model. Organize the data using a matrix. Then write a new matrix giving the fuel economy figures for next year s models if each measure of fuel economy increases by 8%. Economy car: Mid-size car: Mini-van: SUV: 3 mpg in city driving, 40 mpg in highway driving 4 mpg in city driving, 34 mpg in highway driving 18 mpg in city driving, 5 mpg in highway driving 19 mpg in city driving, mpg in highway driving 33. TAKS REASONING In a certain city, an electronics chain has a downtown store and a store in the mall. Each store carries three models of digital camera. Sales of the cameras for May and June are shown. May Downtown sales: 31 of model A, 4 of model B, 18 of model C Mall sales: of model A, 5 of model B, 11 of model C June Downtown sales: 5 of model A, 36 of model B, 1 of model C Mall sales: 38 of model A, 3 of model B, 15 of model C a. Organize the information using two matrices M and J that represent the sales for May and June, respectively. b. Find M 1 J and describe what this matrix sum represents. c. Write a matrix giving the average monthly sales for the two month period. 34. TAKS REASONING The matrices below show the numbers of female athletes who participated in selected NCAA sports and the average team size for each sport during the and seasons. Does the matrix A 1 B give meaningful information? Explain. Basketball Gymnastics Skiing Soccer (A) (B) Athletes Team size Athletes Team size 314, , , Basketball Gymnastics Skiing Soccer 314, , , WORKED-OUT SOLUTIONS on p. WS1 5 TAKS PRACTICE AND REASONING

7 35. CHALLENGE A rectangle has vertices (1, 1), (1, 4), (5, 1), and (5, 4). Write a 3 4 matrix A whose columns are the vertices of the rectangle. Multiply matrix A by 3. In the same coordinate plane, draw the rectangles represented by the matrices A and 3A. How are the rectangles related? MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Skills Review Handbook p. 1004; TAKS Workbook 36. TAKS PRACTICE A health teacher surveyed 100 students to determine their favorite exercise activity or combination of exercise activities. The results are shown at the right. How many of the students surveyed chose only running as their favorite exercise activity? TAKS Obj. 10 A 13 B 9 C 4 D 71 running 15 bicycling swimming REVIEW Lesson.3; TAKS Workbook 37. TAKS PRACTICE Which statement best describes the effect on the graph shown when the y-intercept is decreased by 4? TAKS Obj. 3 F The x-intercept decreases. G The slope decreases. H The x-intercept increases. J The slope increases x 3 4 y QUIZ for Lessons Graph the system of inequalities. (p. 168) 1. y < 6. x 1 3. x 1 3y > 3 x 1 y > x 1 y 5 x 1 3y < 9 4. x y 4 5. x 1 y y < x x 1 4y 10 y x 1 y < 5x 1 9 Solve the system using any algebraic method. (p. 178) 7. x y 3z x 1 y 1 z x 4y 1 3z 5 1 x 1 y 5z 511 x 3y 1 z 5 9 6x 1 y 1 10z 5 19 x 3y x 5y 1 z 5 16 x 1 5y z 5 Use matrices A, B, and C to evaluate the matrix expression, if possible. If not possible, state the reason. (p. 187) A 5F 5 3 1G B 5F C 8 10G 5F 1 4 1G 10. A 1 B 11. B A 1. 3A 1 C APPLES You have $5 to spend on 1 pounds of three types of apples. Empire apples cost $1.40 per pound, Red Delicious apples cost $1.10 per pound, and Golden Delicious apples cost $1.30 per pound. You want twice as many Red Delicious apples as the other two kinds combined. Use a system of equations to find how many pounds of each type you should buy. (p. 178) } 3 C EXTRA PRACTICE for Lesson 3.5, p. 101 ONLINE QUIZ at classzone.com 193

8 Graphing Calculator ACTIVITY Use after Lesson Use Matrix Operations TEKS a.1, a.5, a.6 TEXAS classzone.com Keystrokes QUESTION How can you use a graphing calculator to perform matrix operations? E XAMPLE Perform operations with matrices Using matrices A and B below, find A 1 B and 3A B. A 5F 8 1 B 3 7 9G 5F G STEP 1 Enter matrix A Enter the dimensions and elements of matrix A. STEP Enter matrix B Enter the dimensions and elements of matrix B. STEP 3 Perform calculations From the home screen, calculate A 1 B and 3A B. MATRIX[A] 33 [8-1 ] [3-7 9],3=9 MATRIX[B] 33 [1 0-5] [ ],3=10 [A]+[B] [[9-1 -3] [ ]] 3[A]-[B] [[ -3 16] [ ]] PRACTICE Use a graphing calculator to perform the indicated operation(s). 1. F G 1F. 3 6G.6F G F G1F G 3F GF5 4 3G 5. BOOK SALES The matrices below show book sales (in thousands of dollars) at a chain of bookstores for July and August. The book formats are hardcover and paperback. The categories of books are romance (R), mystery (M), science fiction (S), and children s (C). Find the total sales of each format and category for July and August. July August R M S C R M S C Hardcover Paperback F G F G 194 Chapter 3 Linear Systems and Matrices