3.8. F d 2b provided ad 2 cb Þ 0. F d 2b 5 F Use Inverse Matrices to Solve Linear Systems. For Your Notebook E XAMPLE 1

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1 TEKS 3.8 2A.2.A, 2A.3.A, 2A.3.B, 2A.3.C Use Inverse Matrices to Solve Linear Systems Before You solved linear systems using Cramer s rule. Now You will solve linear systems using inverse matrices. Why? So you can find how many batches of a recipe to make, as in Ex. 45. Key Vocabulary identity matrix inverse matrices matrix of variables matrix of constants The n 3 n identity matrix is a matrix with s on the main diagonal and 0 s elsewhere. If A is any n 3 n matrix and I is the n 3 n identity matrix, then AI 5 A and IA 5 A Identity Matrix Identity Matrix I I G G 0 0 Two n 3 n matrices A and B are inverses of each other if their product (in both orders) is the n 3 n identity matrix. That is, AB 5 I and BA 5 I. An n 3 n matrix A has an inverse if and only if det A Þ 0. The symbol for the inverse of A is A 2. KEY CONCEPT or Your Notebook The Inverse of a Matrix The inverse of the matrix A 5 a c dg b is A 2 5} A d 2b 5 } 2c ag ad 2 cb d 2b provided ad 2 cb Þ 0. 2c ag E XAMPLE ind the inverse of a matrix CHECK INVERSES In Example, you can check the inverse by showing that AA 2 5 I 5 A 2 A. ind the inverse of A G. A 2 5 } G G G GUIDED PRACTICE for Example ind the inverse of the matrix G G 2 22G 20 Chapter 3 Linear Systems and Matrices

2 E XAMPLE 2 Solve a matrix equation Solve the matrix equation AX 5 B for the matrix X. Solution A B GX G Begin by finding the inverse of A. A 2 5} G G 4 7 2G GX G 22 3 A 2 22G 2 AX 5 A 2 B 0 0 GX IX 5 A 3 2G 2 B To solve the equation for X, multiply both sides of the equation by A 2 on the left. at classzone.com X G X 5 A 2 B GUIDED PRACTICE for Example 2 4. Solve the matrix equation GX G. INVERSE O A MATRIX The inverse of a matrix is difficult to compute by hand. A calculator that will compute inverse matrices is useful in this case. E XAMPLE 3 Use a graphing calculator to find the inverse of A. Then use the calculator to verify your result. Solution ind the inverse of a matrix A G Enter matrix A into a graphing calculator and calculate A 2. Then compute AA 2 and A 2 A to verify that you obtain the identity matrix. [A]- [[2-7 3 ] [ ] [.5 -.5]] [A][A]- [[ 0 0] [0 0] [0 0 ]] [A]-[A] [[ 0 0] [0 0] [0 0 ]] 3.8 Use Inverse Matrices to Solve Linear Systems 2

3 GUIDED PRACTICE for Example 3 Use a graphing calculator to find the inverse of the matrix A. Check the result by showing 5 that AA 2 5 I and A 2 A 5 I A A G 2G 8G A KEY CONCEPT or Your Notebook Using an Inverse Matrix to Solve a Linear System STEP Write the system as a matrix equation AX 5 B. The matrix A is the coefficient matrix, X is the matrix of variables, and B is the matrix of constants. STEP 2 ind the inverse of matrix A. STEP 3 Multiply each side of AX 5 B by A 2 on the left to find the solution X 5 A 2 B. E XAMPLE 4 Solve a linear system Use an inverse matrix to solve the linear system. 2x 2 3y 5 9 Equation x 4y 527 Equation 2 SOLVE SYSTEMS You can use the method shown in Example 4 if A has an inverse. If A does not have an inverse, then the system has either no solution or infinitely many solutions. Solution STEP Write the linear system as a matrix equation AX 5 B. coefficient matrix of matrix of matrix (A) variables (X) constants (B) 2 23 p 4G yg x 5 27G 9 STEP 2 ind the inverse of matrix A. A 2 5 } 8 2 (23) G } 2} 2 }4 STEP 3 Multiply the matrix of constants by A 2 on the left. 4 } 3 } 53 X 5 A 2 B 2} 2 }4 9 27G5 5 23G 5 x yg c The solution of the system is (5, 23). CHECK 2(5) 2 3(23) (23) at classzone.com 22 Chapter 3 Linear Systems and Matrices

4 E XAMPLE 5 TAKS REASONING: Multi-Step Problem GITS A company sells three types of movie gift baskets. A basic basket with 2 movie passes and package of microwave popcorn costs $5.50. A medium basket with 2 movie passes, 2 packages of popcorn, and DVD costs $37. A super basket with 4 movie passes, 3 packages of popcorn, and 2 DVDs costs $ ind the cost of each item in the gift baskets. ANOTHER WAY or an alternative method for solving the problem in Example 5, turn to page 28 for the Problem Solving Workshop. Solution STEP Write verbal models for the situation. 2 p 2 p movie pass movie pass 2 p popcorn 5 popcorn basic basket DVD 5 medium basket Equation Equation 2 4 p movie pass 3 p popcorn 2 p DVD 5 super basket Equation 3 STEP 2 STEP 3 Write a system of equations. Let m be the cost of a movie pass, p be the cost of a package of popcorn, and d be the cost of a DVD. 2m p Equation 2m 2p d Equation 2 4m 3p 2d Equation 3 Rewrite the system as a matrix equation p dg G 4 3 2Gm STEP 4 Enter the coefficient matrix A and the matrix of constants B into a graphing calculator. Then find the solution X 5 A 2 B. MATRIX[A] 333 [2 0] [2 2 ] [4 3 2] MATRIX[B] 33 [5.5] [37 ] [72.5] [A]-[B] [[7 ] [.5] [20 ]] 3,3=2 3,=72.5 c A movie pass costs $7, a package of popcorn costs $.50, and a DVD costs $20. GUIDED PRACTICE for Examples 4 and 5 Use an inverse matrix to solve the linear system. 8. 4x y x 2 y x 2 y 525 3x 5y 52 6x 2 3y x 2y 5 8. WHAT I In Example 5, how does the answer change if a basic basket costs $7, a medium basket costs $35, and a super basket costs $69? 3.8 Use Inverse Matrices to Solve Linear Systems 23

5 3.8 EXERCISES HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Exs. 3, 25, and 47 5 TAKS PRACTICE AND REASONING Exs. 2, 34, 4, 46, 5, 52, and 53 5 MULTIPLE REPRESENTATIONS Ex. 45 SKILL PRACTICE. VOCABULARY Identify the matrix of variables and the matrix of constants in the matrix equation GyG 5 22G x 4 2. WRITING Explain how to find the inverse of a matrix A where det A Þ 0. EXAMPLE on p. 20 for Exs. 3 2 EXAMPLE 2 on p. 2 for Exs. 3 8 EXAMPLE 3 on p. 2 for Exs INDING INVERSES ind the inverse of the matrix G G G ERROR ANALYSIS Describe and correct the error in finding the inverse of the matrix 2 4 5G G 2. TAKS REASONING What is the inverse of the matrix G 2 3G 26 30G G? A G B G C 23 SOLVING EQUATIONS Solve the matrix equation. 4 5 }3 }6 24 2G 0 23 D 3 20G 3 2G GX G GX G GX G GX G GX G GX INDING INVERSES Use a graphing calculator to find the inverse of matrix A. Check the result by showing that AA 2 5 I and A 2 A 5 I. 5 0G 4G A A A 3 4 0G A G G G 2G A A G 5 6G G 24 Chapter 3 Linear Systems and Matrices

6 EXAMPLE 4 on p. 22 for Exs SYSTEMS O TWO EQUATIONS Use an inverse matrix to solve the linear system x 2 y x 7y x 2 2y x 2 2y x 3y 524 6x 2 5y x 2 y x 2 9y x 2 7y 526 9x 2 0y x 6y 5 8 2x 5y x y x y x 7y x 3y x 5y x 5y TAKS REASONING What is the solution 3x 2 5y 5226 of the system shown? 2x 2y 5 0 A (3, 7) B (7, 2) C (22, 4) D (68, 0) EXAMPLE 5 on p. 23 for Exs SYSTEMS O THREE EQUATIONS Use an inverse matrix and a graphing calculator to solve the linear system. 35. x 2 y 2 3z x y 2 8z x 4y 5z 5 5 5x 2y z 527 x 2 2y z 52 x 2y 3z x 2 y 5 8 2x 2 2y 5z 527 5x 2 4y 2 2z x 2 y 2 z x 2y 2 z x y 2z 5 6x 2 z x 2 5y 4z 5248 x 2 y z 525 2x 4y 5z x y z 5 2 2x 4y 2 z TAKS REASONING Write a matrix that has no inverse. 42. CHALLENGE Solve the linear system using the given inverse of the coefficient matrix. G 2w 5x 2 4y 6z x y 2 7z w 8x 2 7y 4z 5225 A w 6x 2 5y 0z PROBLEM SOLVING EXAMPLES 4 and 5 on pp for Exs AVIATION A pilot has 200 hours of flight time in single-engine airplanes and twin-engine airplanes. Renting a single-engine airplane costs $60 per hour, and renting a twin-engine airplane costs $240 per hour. The pilot has spent $2,000 on airplane rentals. Use an inverse matrix to find how many hours the pilot has flown each type of airplane. 44. BASKETBALL During the NBA season, Dirk Nowitzki of the Dallas Mavericks made a total of 976 shots and scored 680 points. His shots consisted of 3-point field goals, 2-point field goals, and -point free throws. He made 35 more 2-point field goals than free throws. Use an inverse matrix to find how many of each type of shot he made. 3.8 Use Inverse Matrices to Solve Linear Systems 25

7 45. MULTIPLE REPRESENTATIONS A cooking class wants to use up 8 cups of buttermilk and eggs by baking rolls and muffins to freeze. A batch of rolls uses 2 cups of buttermilk and 3 eggs. A batch of muffins uses cup of buttermilk and egg. a. Writing a System Write a system of equations for this situation. b. Writing a Matrix Equation Write the system of equations from part (a) as a matrix equation AX 5 B. c. Solving a System Use an inverse matrix to solve the system of equations. How many batches of each recipe should the class make? 46. TAKS REASONING A company sells party platters with varying assortments of meats and cheeses. A basic platter with 2 cheeses and 3 meats costs $8, a medium platter with 3 cheeses and 5 meats costs $28, and a super platter with 7 cheeses and 0 meats costs $60. a. Write and solve a system of equations using the information about the basic platter and the medium platter. b. Write and solve a system of equations using the information about the medium platter and the super platter. c. Compare the results from parts (a) and (b) and make a conjecture about why there is a discrepancy. 47. NUTRITION The table shows the calories, fat, and carbohydrates per ounce for three brands of cereal. How many ounces of each brand should be combined to get 500 calories, 3 grams of fat, and 00 grams of carbohydrates? Round your answers to the nearest tenth of an ounce. Cereal Calories at Carbohydrates Bran Crunchies 78 g 22 g Toasted Oats 04 0 g 25.5 g Whole Wheat lakes g 23.8 g 48. MULTI-STEP PROBLEM You need 9 square feet of glass mosaic tiles to decorate a wall of your kitchen. You want the area of the red tiles to equal the combined area of the yellow and blue tiles. The cost of a sheet of glass tiles having an area of 0.75 square foot is $6.50 for red, $4.50 for yellow, and $8.50 for blue. You have $80 to spend. a. Write a system of equations to represent this situation. b. Rewrite the system as a matrix equation. c. Use an inverse matrix to find how many sheets of each color tile you should buy. 49. GEOMETRY The columns of matrix T below give the coordinates of the vertices of a triangle. Matrix A is a transformation matrix. A 5 0 T 2 0G G a. ind AT and AAT. Then draw the original triangle and the two transformed triangles. What transformation does A represent? b. Describe how to use matrices to obtain the original triangle represented by T from the transformed triangle represented by AAT. Mosaic tiles 26 5 WORKED-OUT SOLUTIONS on p. WS 5 TAKS PRACTICE AND REASONING 5 MULTIPLE REPRESENTATIONS

8 50. CHALLENGE Verify the formula on page 20 for the inverse of a matrix by showing that AB 5 I and BA 5 I for the matrices A and B given below. A 5 a b B 5 } c dg ad 2 cb d 2b 2c ag MIXED REVIEW OR TAKS TAKS PRACTICE at classzone.com REVIEW TAKS Preparation p. 66; TAKS Workbook REVIEW Lesson.3; TAKS Workbook REVIEW Lesson 3.2; TAKS Workbook 5. TAKS PRACTICE A grocer wants to mix peanuts worth $2.50 per pound with 2 pounds of cashews worth $4.75 per pound. To obtain a nut mixture worth $4 per pound, how many pounds of peanuts are needed? TAKS Obj. 0 A 3.6 lb B 6 lb C 2 lb D 8 lb 52. TAKS PRACTICE The sum of three numbers is 4. The second number is 5 less than three times the first number. The third number is 2 more than four times the first number. Which equation represents the relationship between the three numbers where n is the first number? TAKS Obj. 4 5 n 2 (3n 2 5) 2 (4n 2) G 4 5 n (4n 2 5) (3n 2) H 4 5 n (3n 2 5) (4n 2) J 4 5 n (5 2 3n) (2 4n) 53. TAKS PRACTICE Which ordered pair is the solution of this system of linear equations? TAKS Obj. 4 5x y 527 2x 2 7y 5 8 A (23, 22) B (23, 2) C 3, 2 2 } 7 2 D (, 2) QUIZ for Lessons Using the given matrices, evaluate the expression. (p. 95) A 5 24 B 5 2G, C 0 G, 5 2 4G. 2AB 2. AB AC 3. A(B C) 4. (B 2 A)C Evaluate the determinant of the matrix. (p. 203) G G 7. Use an inverse matrix to solve the linear system. (p. 20) 2 G x 3y x 2 4y x 2y 523 2x 7y 526 2x 2 3y 5 3 6x 2 5y x 2 y x 4y x y 522 2x 2 2y 528 5x 3y x y BOATING You are making a triangular sail for a sailboat. The vertices of the sail are (0, 2), (2, 2), and (2, 26) where the coordinates are measured in feet. ind the area of the sail. (p. 203) EXTRA PRACTICE for Lesson 3.8, p. 02 ONLINE QUIZ at classzone.com 27

9 LESSON 3.8 TEKS 2A.3.A, 2A.3.B, 2A.3.C Using ALTERNATIVE METHODS Another Way to Solve Example 5, page 23 MULTIPLE REPRESENTATIONS In Example 5 on page 23, you solved a linear system using an inverse matrix. You can also solve systems using augmented matrices. An augmented matrix for a system contains the system s coefficient matrix and matrix of constants. Linear System x 2 4y x 7y 522 Augmented Matrix G Recall from Lesson 3.2 that an equation in a system can be multiplied by a constant, or a multiple of one equation can be added to another equation. Similar operations can be performed on the rows of an augmented matrix to solve the corresponding system. KEY CONCEPT or Your Notebook Elementary Row Operations for Augmented Matrices Two augmented matrices are row-equivalent if their corresponding systems have the same solution(s). Any of these row operations performed on an augmented matrix will produce a matrix that is row-equivalent to the original: Interchange two rows. Multiply a row by a nonzero constant. Add a multiple of one row to another row. P ROBLEM GITS A company sells three types of movie gift baskets. A basic basket with 2 movie passes and package of microwave popcorn costs $5.50. A medium basket with 2 movie passes, 2 packages of popcorn, and DVD costs $37. A super basket with 4 movie passes, 3 packages of popcorn, and 2 DVDs costs $ ind the cost of each item in the gift baskets. M ETHOD Using an Augmented Matrix You need to write a linear system, write the corresponding augmented matrix, and use row operations to transform the augmented matrix into a matrix with s along the main diagonal and 0 s below the main diagonal. Such a matrix is in triangular form and can be used to solve for the variables in the system. Let m be the cost of a movie pass, p be the cost of a package of popcorn, and d be the cost of a DVD. 28 Chapter 3 Linear Systems and Matrices

10 STEP Write a linear system and then write an augmented matrix. 2m p m 2p d m 3p 2d G STEP 2 Add 22 times the first row to the third row (22)R R G STEP 3 Add 2 times the first row to the second row (2)R R G STEP 4 Add 2 times the second row to the third row (2)R 2 R G STEP 5 Multiply the first row by R G The third row of the matrix tells you that d Substitute 20 for d in the equation for the second row, p d 5 2.5, to obtain p , or p 5.5. Then substitute.5 for p in the equation for the first row, m 0.5p , to obtain m 0.5(.5) , or m 5 7. c A movie pass costs $7, a package of popcorn costs $.50, and a DVD costs $20. P RACTICE. WHAT I? In the problem on page 28, suppose a basic basket costs $7.75, a medium basket costs $34.50, and a super basket costs $ Use an augmented matrix to find the cost of each item. 2. INANCE You have $8,000 to invest. You want an overall annual return of 8%. The expected annual returns are 0% for a stock fund, 7% for a bond fund, and 5% for a money market fund. You want to invest as much in stocks as in bonds and the money market combined. Use an augmented matrix to find how much to invest in each fund. 3. BIRDSEED A pet store sells 20 pounds of birdseed for $0.85. The birdseed is made from two kinds of seeds, sunflower seeds and thistle seeds. Sunflower seeds cost $.34 per pound and thistle seeds cost $.79 per pound. Use an augmented matrix to find how many pounds of each variety are in the mixture. 4. REASONING Solve the given system using an augmented matrix. What can you say about the system s solution(s)? x 2 2y 4z 520 5x y 2 z x 2 6y 2z 5230 Using Alternative Methods 29