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z z 2w w w 2 3 2 3. Using back-substitution, the solution is x 1, y 2, z 1, and w 3. Now try Exercise 51. 333202_0801.qxd 12/5/05 10:59 AM Page 578 578 Chapter 8 Matrices and Determinants The procedure for using Gaussian elimination with back-substitution is summarized below. Gaussian Elimination with Back-Substitution 1. Write the augmented matrix of the system of linear equations. 2. Use elementary row operations to rewrite the augmented matrix in row-echelon form. 3. Write the system of linear equations corresponding to the matrix in row-echelon form, and use back-substitution to find the solution. When solving a system of linear equations, remember that it is possible for the system to have no solution. If, in the elimination process, you obtain a row with zeros except for the last entry, it is unnecessary to continue the elimination process. You can simply conclude that the system has no solution, or is inconsistent. Example 7 A System with No Solution Solve the system x x 2x 3x y 2z 4 z 6 3y 5z 4 2y z 1. Solution R1 2R1 3R1 R2 R3 R4 R2 R3 1 2 3 1 1 →.................................... 4 6 4 1 4 2 4 11 4 2 2 11 Write augmented matrix. Perform row operations. Perform row operations. Note that the third row of this matrix consists of zeros except for the last entry. This means that the original system of linear equations is inconsistent. You can see why this is true by converting back to a system of linear equations. x y 2z y z 0 5y 7z 4 2 2 11 Because the third equation is not possible, the system has no solution. Now try Exercise 57. 333202_0801.qxd 12/5/05 10:59 AM Page 579 Te c h n o l o g y matrix, see For a demonstration of a graphical approach to Gauss-Jordan elimi2 3 nation on a the Visualizing Row Operations Program available for several models of graphing calculators at our website college.hmco.com. The advantage of using GaussJordan elimination to solve a system of linear equations is that the solution of the system is |
easily found without using back-substitution, as illustrated in Example 8. Section 8.1 Matrices and Systems of Equations 579 Gauss-Jordan Elimination With Gaussian elimination, elementary row operations are applied to a matrix to obtain a (row-equivalent) row-echelon form of the matrix. A second method of elimination, called Gauss-Jordan elimination, after Carl Friedrich Gauss and Wilhelm Jordan (1842–1899), continues the reduction process until a reduced row-echelon form is obtained. This procedure is demonstrated in Example 8. Example 8 Gauss-Jordan Elimination Use Gauss-Jordan elimination to solve the system x 2y x 3y 2x 5y 3z 5z 9 4 17. Solution In Example 4, Gaussian elimination was used to obtain the row-echelon form of the linear system above.... 2... 1... Now, apply elementary row operations until you obtain zeros above each of the leading 1’s, as follows. 2R2 R1 9R3 3R3 R1 R2 0 0 →1 →.................. 19 5 2 1 1 2 Perform operations on R1 so second column has a zero above its leading 1. Perform operations on R1 and R2 so third column has zeros above its leading 1. The matrix is now in reduced row-echelon form. Converting back to a system of linear equations, you have x y z 1 1. 2 Now you can simply read the solution, 1, 1, 2. written as the ordered triple Now try Exercise 59. x 1, y 1, and z 2, which can be The elimination procedures described in this section sometimes result in fractional coefficients. For instance, in the elimination procedure for the system 2x 5y 3x 2y 3x 3y 5z 3z 17 11 6 you may be inclined to multiply the first row by to produce a leading 1, which will result in working with fractional coefficients. You can sometimes avoid fractions by judiciously choosing the order in which you apply elementary row operations. 1 2 333202_0801.qxd 12/5/05 10:59 AM Page 580 580 Chapter 8 Matrices and Determinants Recall from Chapter 7 that when there are fewer equations than variables in a system of equations, then the system has either no solution or infinitely many solutions. Example 9 A System with an Infinite Number |
of Solutions Solve the system. 2x 4y 3x 5y 2z 0 1 Solution 2 3 →1 3 0 →1 →1 0 → 2R1 3R1 R2 R2 R1 2R2.............................. The corresponding system of equations is x 5z 2. y 3z 1 x y Solving for and x 5z 2 z, in terms of you have y 3z 1. and To write a solution to the system that does not use any of the three variables of the system, let z a. represent any real number and let a y x In Example 9, and are solved for in terms of the third variable z. To write a solution to the system that does not use any of the three variables of the system, a let represent any real number x and let and The solution can then be written in terms of which is not one of the variables of the system. Then solve for z a. a, y. in the equations for and x y. Now substitute z a for x 5z 2 5a 2 y 3z 1 3a 1 So, the solution set can be written as an ordered triple with the form 5a 2, 3a 1, a where a is any real number. Remember that a solution set of this form represents a an infinite number of solutions. Try substituting values for to obtain a few solutions. Then check each solution in the original equation. Now try Exercise 65. It is worth noting that the row-echelon form of a matrix is not unique. That is, two different sequences of elementary row operations may yield different row-echelon forms. This is demonstrated in Example 10. 333202_0801.qxd 12/5/05 10:59 AM Page 581 Section 8.1 Matrices and Systems of Equations 581 Example 10 Comparing Row-Echelon Forms Compare the following row-echelon form with the one found in Example 4. Is it the same? Does it yield the same solution? x 2y x 3y 2x 5y 3z 5z R2 1 2 1 2 1 2 1 R1 R1 R1 2R1 R2 R3 R2 R3 1 2R3 9 4 17................................................ |
...... 17 4 9 17 4 9 17 4 5 9 4 5 4 4 5 2 Solution This row-echelon form is different from that obtained in Example 4. The corresponding system of linear equations for this row-echelon matrix is x 3y y 4 3z 5. z 2 Using back-substitution on this system, you obtain the solution x 1, y 1, and z 2 which is the same solution that was obtained in Example 4. Now try Exercise 77. You have seen that the row-echelon form of a given matrix is not unique; however, the reduced row-echelon form of a given matrix is unique. Try applying Gauss-Jordan elimination to the row-echelon matrix in Example 10 to see that you obtain the same reduced row-echelon form as in Example 8. 333202_0801.qxd 12/5/05 10:59 AM Page 582 582 Chapter 8 Matrices and Determinants 8.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. VOCABULARY CHECK: Fill in the blanks. 1. A rectangular array of real numbers than can be used to solve a system of linear equations is called a ________. 2. A matrix is ________ if the number of rows equals the number of columns. 3. For a square matrix, the entries a11, a22, a33,..., ann are the ________ ________ entries. 4. A matrix with only one row is called a ________ matrix and a matrix with only one column is called a ________ matrix. 5. The matrix derived from a system of linear equations is called the ________ matrix of the system. 6. The matrix derived from the coefficients of a system of linear equations is called the ________ matrix of the system. 7. Two matrices are called ________ if one of the matrices can be obtained from the other by a sequence of elementary row operations. 8. A matrix in row-echelon form is in ________ ________ ________ if every column that has a leading 1 has zeros in every position above and below its leading 1. 9. The process of using row operations to write a matrix in reduced row-echelon form is called ________ ________. PR |
EREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 6, determine the order of the matrix. 1. 7 0 2 36 3 33 9 3. 5. 45 20 2. 4. 6 15 3 6 7 0 3 7 In Exercises 7–12, write the augmented matrix for the system of linear equations. 7. 9. 11. 3y 3y 4x x x 10y 5x 3y 2x y 7x 5 12 2z 2 4z 0 6 5y z 13 8z 10 19x 8. 10. 12. 7x 4y 22 5x 9y 15 x 8y 9x 5z 15z 8z 2y 3z 25y 11z 7x 3x y 8 38 20 20 5 In Exercises 13–18, write the system of linear equations y, represented by the augmented matrix. (Use variables z, w, x, and 1 2 7 8 2 0 6 13. 14. 15. 7 4 2 3 5 3 if applicable.) 0 2 5 2 0 0 1 3 12 7 2 16. 17. 18. 4 11 12 18 10 1 0 2 0 0 5 3 6 11 18 25 29 0 10 4 10 25 7 23 21 In Exercises 19–22, fill in the blank(s) using elementary row operations to form a row-equivalent matrix. 19. 21 10 4 1 8 1 3 5 3 1 4 10 12 1 3 6 20. 22 333202_0801.qxd 12/5/05 10:59 AM Page 583 In Exercises 23–26, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix. Original Matrix 2 3 5 1 8 1 Original Matrix 4 1 3 4 7 3 Original Matrix 1 5 7 3 5 1 Original Matrix New Row-Equivalent Matrix 39 13 8 3 New Row-Equivalent Matrix 3 5 1 0 4 5 1 New Row-Equivalent Matrix 7 5 27 New Row-Equivalent Matrix 3 1 7 6 5 27 11 4 5 6 3 2 7 6 23. 24. 25. 26. 27. Perform the sequence of row operations on the matrix. What did the operations accomplisha) Add (b) Add 2 3 1 (d) Multiply 2 (c) Add times times times R2 times by to to R1 R1 R2 to 1 |
5. to R2. R3. R3. (e) Add R1. 28. Perform the sequence of row operations on the matrix. R2 What did the operations accomplish? 7 0 3 4 1 2 4 1 to R3 (a) Add (c) Add 3 times R4. R1 (b) Interchange R1 times R2 7 (e) Multiply (d) Add by and R4. R3. to R4. to R1 1 2. (f) Add the appropriate multiples of R2 to R1, R3, and R4. In Exercises 29–32, determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. 29 Section 8.1 Matrices and Systems of Equations 583 30. 31. 32 10 0 In Exercises 33–36, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.) 33. 35 18 5 10 14 1 8 0 34. 36 10 10 1 5 3 0 1 2 3 14 8 7 23 24 In Exercises 37–42, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form 37. 39. 40. 41. 42. 38. 1 5 2 3 15 6 2 9 10 11 1 5 2 10 1 1 2 10 5 9 3 14 2 8 0 30 12 4 4 32 z, y, x, 43. and In Exercises 43–46, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve. (Use variables if applicable.) 1 0 1 46 44. 45. 5 1 0 0 0 0 333202_0801.qxd 12/5/05 11:00 AM Page 584 584 Chapter 8 Matrices and Determinants In Exercises 47–50, an augmented matrix that represents a system of linear equations (in variables if applicable) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. and x, y, z, 47. 48. 49. 50 10 4 10 4 5 3 0 In Exercises 51–70, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. |
4 34 5 7 5 10 3z 24 z 14 6 2 28 14 2x 6y 16 2x 3y 7 x y 2x 4y 5x 5y 2x 3y x 3y 2x 6y 2x y 2x 3x 2x 2y 7x 5y 2y 3y y x x x x z z 4x 8x 15 10 14 z 2y 2z y 4z y 3z 3 3y 7z 5 9y 15z 9 51. 53. 55. 57. 59. 61. 63. 65. 67. 68. 69. 52. 54. 56. 58. 60. 62. 64. 66. 1.5 z z z 2x 3x 3x 2x y y 2y y 2y 27 13 22 9 3 3z 2z z 2 5 4 14 21 19 28 0 5 x 2y 7 2x y 8 3x 2y x 3y 2x 6y x 2y x 2y 2x 4y x x x 2y 4y y x 5z 3 2z 1 z 0 x 2y z 2w 8 3x 7y 6z 9w 26 4x 12y 7z 20w 22 3x 9y 5z 28w 30 x y 22 4 32 3x 4y 4x 8y 3z 2z z x 2x y y x 70. In Exercises 71–76, use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system. 71. 72. 73. 74. 75. 76. 3x 4y x 5y 5x 2y x 5y 2z x 5y z 3x 15y 3z x y 4z 2 2x 5y 20z 10 x 2y 8z 4 6 6 3 9 z 2w w 2z 6w z w 3x 3y 12z 6 2x 10y 2z 2x y x 2y 2z 4w x x 2y 3x 6y 5z 12w x 3y 3z 2w 6x y z w z w 0 z 2w 0 z 0 z 3w 0 w 0 z 2w 0 x y y y 3y 5y 2x 3x 6 1 3 3 11 30 5 9 In Exercises 77–80, determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices. 77. (a) |
78. (a) 79. (a) 80. (a) y z z y 5z z x 2y z x 3y 4z x 4y 5z x 3y z y 7z z y 6z z (b) (b) (b) (b) 6 16 3 11 4 2 27 54 8 19 18 4 y 6 8 3 11 4 2 y 3z z 3z z x y 2z x 4y x 6y z 15 x y 3z y 5z 42 z 8 15 14 4 y 2z z x 2y 0 x y 6 3x 2y 8 81. Use the system x 3y z 3 x 5y 5z 1 2x 6y 3z 8 to write two different matrices in row-echelon form that yield the same solution. 333202_0801.qxd 12/5/05 11:00 AM Page 585 82. Electrical Network The currents in an electrical network are given by the solution of the system I1 3I1 I2 4I2 I2 I3 3I3 0 18 6 Section 8.1 Matrices and Systems of Equations 585 89. Mathematical Modeling A videotape of the path of a ball thrown by a baseball player was analyzed with a grid covering the TV screen. The tape was paused three times, and the position of the ball was measured each time. The are coordinates obtained are shown in the table. ( and measured in feet.) x y I1, I2, I3 where system of equations using matrices. and are measured in amperes. Solve the 83. Partial Fractions Use a system of equations to write the partial fraction decomposition of the rational expression. Solve the system using matrices. 4x 2 x 12x 1 A x 1 B x 1 C x 12 84. Partial Fractions Use a system of equations to write the partial fraction decomposition of the rational expression. Solve the system using matrices. 8x2 x 12x 1 A x 1 B x 1 C x 12 85. Finance A small shoe corporation borrowed $1,500,000 to expand its line of shoes. Some of the money was borrowed at 7%, some at 8%, and some at 10%. Use a system of equations to determine how much was borrowed at each rate if the annual interest was $130,500 and the amount borrowed at 10% was 4 times the amount borrowed at 7%. Solve the system using matrices. |
86. Finance A small software corporation borrowed $500,000 to expand its software line. Some of the money was borrowed at 9%, some at 10%, and some at 12%. Use a system of equations to determine how much was borrowed at each rate if the annual interest was $52,000 and the amount borrowed at 10% was times the amount borrowed at 9%. Solve the system using matrices. 21 2 In Exercises 87 and 88, use a system of equations to find the specified equation that passes through the points. Solve the system using matrices. Use a graphing utility to verify your results. 87. Parabola: 88. Parabola: y ax 2 bx c y 24 (3, 20) (2, 13) (1, 8) −8 −4 4 8 12 x y ax 2 bx c y 12 8 (1, 9) (2, 8) (3, 5) −8 −4 8 12 x Horizontal distance, x Height, y 0 15 30 5.0 9.6 12.4 (a) Use a system of equations to find the equation of the that passes through the parabola three points. Solve the system using matrices. y ax 2 bx c (b) Use a graphing utility to graph the parabola. (c) Graphically approximate the maximum height of the ball and the point at which the ball struck the ground. (d) Analytically find the maximum height of the ball and the point at which the ball struck the ground. (e) Compare your results from parts (c) and (d). Model It 90. Data Analysis: Snowboarders The table shows the numbers of people (in millions) in the United States who participated in snowboarding for selected years from 1997 to 2001. (Source: National Sporting Goods Association) y Year Number, y 1997 1999 2001 2.8 3.3 5.3 (a) Use a system of equations to find the equation of y at2 bt c that passes through the parabola t 7 the points. Let corresponding to 1997. Solve the system using matrices. represent the year, with t (b) Use a graphing utility to graph the parabola. (c) Use the equation in part (a) to estimate the number of people who participated in snowboarding in 2003. How does this value compare with the actual 2003 value of 6.3 million? y (d) Use the |
equation in part (a) to estimate in the year 2008. Is the estimate reasonable? Explain. 333202_0801.qxd 12/5/05 11:00 AM Page 586 586 Chapter 8 Matrices and Determinants Network Analysis In Exercises 91 and 92, answer the questions about the specified network. (In a network it is assumed that the total flow into each junction is equal to the total flow out of each junction.) 91. Water flowing through a network of pipes (in thousands of cubic meters per hour) is shown in the figure. 600 600 x3 x 1 x6 x4 x2 x7 x5 500 500 (a) Solve this system using matrices for the water flow xi, i 1, 2,..., 7. represented by (b) Find the network flow pattern when 0. x 7 0 x6 and (c) Find the network flow pattern when 0. x6 1000 and x5 92. The flow of traffic (in vehicles per hour) through a network of streets is shown in the figure. 300 200 x2 x 1 x5 x3 x 4 150 350 represented by (a) Solve this system using matrices for the traffic flow xi, i 1, 2,..., 5. 200 150 (b) Find the traffic flow when (c) Find the traffic flow when 50. 0. and and x3 x3 x2 x2 Synthesis True or False? statement is true or false. Justify your answer. In Exercises 93–95, determine whether the 93. 5 1 94. The matrix 0 3 2 6 7 0 is a 4 2 matrix is in reduced row-echelon form. 95. The method of Gaussian elimination reduces a matrix until a reduced row-echelon form is obtained. 96. Think About It The augmented matrix represents z and ) that a system of linear equations (in variables has been reduced using Gauss-Jordan elimination. Write a system of equations with nonzero coefficients that is represented by the reduced matrix. (There are many correct answers.) y, x 97. Think About It (a) Describe the row-echelon form of an augmented matrix that corresponds to a system of linear equations that is inconsistent. (b) Describe the row-echelon form of an augmented matrix that corresponds to a system of linear equations that has an infinite number of solutions. 98. Describe the three elementary row operations that |
can be performed on an augmented matrix. 99. What is the relationship between the three elementary row operations performed on an augmented matrix and the operations that lead to equivalent systems of equations? 100. Writing In your own words, describe the difference between a matrix in row-echelon form and a matrix in reduced row-echelon form. Skills Review In Exercises 101–106, sketch the graph of the function. Do not use a graphing utility. fx 2x2 4x 3x x2 fx x2 2x 1 102. 101. x2 1 103. 104. 105. 106. f x 2 x1 gx 3x2 hx lnx 1 f x 3 ln x 333202_0802.qxd 12/5/05 10:57 AM Page 587 8.2 Operations with Matrices Section 8.2 Operations with Matrices 587 What you should learn • Decide whether two matrices are equal. • Add and subtract matrices and multiply matrices by scalars. • Multiply two matrices. • Use matrix operations to model and solve real-life problems. Why you should learn it Matrix operations can be used to model and solve real-life problems. For instance, in Exercise 70 on page 601, matrix operations are used to analyze annual health care costs. Equality of Matrices In Section 8.1, you used matrices to solve systems of linear equations. There is a rich mathematical theory of matrices, and its applications are numerous. This section and the next two introduce some fundamentals of matrix theory. It is standard mathematical convention to represent matrices in any of the following three ways.,., or bij aij brackets, such as Representation of Matrices 1. A matrix can be denoted by an uppercase letter such as A, B, or C. 2. A matrix can be denoted by a representative element enclosed in cij 3. A matrix can be denoted by a rectangular array of numbers such as a13 a23 a33... am3 a11 a12 a22 a32... am2 a1n a2n a3n... amn a21 a31... am1. A aij............ Two matrices and A aij for and 1 ≤ i ≤ m B bij and 1 ≤ j ≤ n. are equal if they have the same order In other words, two matrices m n are |
equal if their corresponding entries are equal. bij aij Example 1 Equality of Matrices Solve for a11 a21 a11, a12, a21, and a12 a22 2 3 in the following matrix equation. a22 1 0 © Royalty-Free/Corbis Solution Because two matrices are equal only if their corresponding entries are equal, you can conclude that a11 2, a12 1, a21 3, and a22 0. Now try Exercise 1. Be sure you see that for two matrices to be equal, they must have the same order and their corresponding entries must be equal. For instance.5 but 333202_0802.qxd 12/5/05 10:57 AM Page 588 588 Chapter 8 Matrices and Determinants Matrix Addition and Scalar Multiplication In this section, three basic matrix operations will be covered. The first two are matrix addition and scalar multiplication. With matrix addition, you can add two matrices (of the same order) by adding their corresponding entries Historical Note Arthur Cayley (1821–1895), a British mathematician, invented matrices around 1858. Cayley was a Cambridge University graduate and a lawyer by profession. His groundbreaking work on matrices was begun as he studied the theory of transformations. Cayley also was instrumental in the development of determinants. Cayley and two American mathematicians, Benjamin Peirce (1809–1880) and his son Charles S. Peirce (1839–1914), are credited with developing “matrix algebra.” Definition of Matrix Addition B bij If m n A aij and matrix given by bij A B aij. are matrices of order m n, their sum is the The sum of two matrices of different orders is undefined. Example 2 Addition of Matrices. 0 1 1 1 c. 3 2 d. The sum of and 1 3 4 is undefined because A is of order 3 3 and B is of order 3 2. Now try Exercise 7(a). In operations with matrices, numbers are usually referred to as scalars. In this text, scalars will always be real numbers. You can multiply a matrix by a scalar by multiplying each entry in by c. A A c Definition of Scalar Multiplication If c A aij m n is the matrix given by. cA caij matrix and m n is an c is a scalar, the scalar multiple |
of by A 333202_0802.qxd 12/5/05 10:57 AM Page 589 Section 8.2 Operations with Matrices 589 Exploration Consider matrices A, B, and C below. Perform the indicated operations and compare the results. Find b. Find B A. and then add C to c. Find the resulting matrix. Find B C, then add A to the resulting matrix. 2B, two resulting matrices. Find A B, resulting matrix by 2. then multiply the and 2A then add the 1, 7 2 6 A B A B, The symbol A Moreover, if 1B. A and 1A. sum of A, represents the negation of which is the scalar product A B and represents the That is, are of the same order, then A B A B A 1B. Subtraction of matrices 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, as shown in Example 3(c). Example 3 Scalar Multiplication and Matrix Subtraction For the following matrices, find (a) 3A, (b) B, and (c and B 2 1 1 0 4 3 3A B. 0 3 2 Solution a. b. c. 9 6 3 2 33 32 3A 3 2 32 6 B 1 2 2 3A 10 7 4 1 2 2 0 1 32 30 31 6 0 3 12 3 6 34 31 32 12 3 6 6 4 0 12 6 4 Scalar multiplication Multiply each entry by 3. Simplify. Definition of negation Multiply each entry by 1. 0 4 3 0 3 2 Matrix subtraction Subtract corresponding entries. Now try Exercises 7(b), (c), and (d). It is often convenient to rewrite the scalar multiple by factoring out of every entry in the matrix. For instance, in the following example, the scalar has been factored out of the matrix. 3 1 2 1 1 21 cA 1 2 1 2 5 c 333202_0802.qxd 12/5/05 10:57 AM Page 590 590 Chapter 8 Matrices and Determinants The properties of matrix addition and scalar multiplication are similar to those of addition and multiplication of real numbers. Properties of Matrix Addition and Scalar Multiplication Let c matrices and let and be scalars. m n d B,A, |
C and be cdA cdA) 1A A cA B cA cB c dA cA dA 1. 2. 3. 4. 5. 6. Commutative Property of Matrix Addition Associative Property of Matrix Addition Associative Property of Scalar Multiplication Scalar Identity Property Distributive Property Distributive Property Note that the Associative Property of Matrix Addition allows you to write expressions such as without ambiguity because the same sum occurs no matter how the matrices are grouped. This same reasoning applies to sums of four or more matrices. A B C Example 4 Addition of More than Two Matrices By adding corresponding entries, you obtain the following sum of four matrices Now try Exercise 13. Example 5 Using the Distributive Property Perform the indicated matrix operations. 4 3 32 4 2 7 0 1 Solution 32 4 4 3 0 1 2 7 32 4 6 12 6 21 Now try Exercise 15. 2 7 6 21 34 0 3 1 12 0 9 3 6 24 In Example 5, you could add the two matrices first and then multiply the matrix by 3, as follows. Notice that you obtain the same result. 6 2 24 7 32 7 6 21 32 4 4 3 2 8 0 1 Te c h n o l o g y Most graphing utilities have the capability of performing matrix operations. Consult the user’s guide for your graphing utility for specific keystrokes. Try using a graphing utility to find the sum of the matrices A 2 1 3 0 and B 1 2. 4 5 333202_0802.qxd 12/5/05 10:57 AM Page 591 Section 8.2 Operations with Matrices 591 One important property of addition of real numbers is that the number 0 is for any real number For matrices, a m n zero O A O A. c 0 c the additive identity. That is, A similar property holds. That is, if is an matrix consisting entirely of zeros, then O matrices. For example, the following matrices are the additive identities for the set of all 2 3 is the additive identity for the set of all In other words, c. is the matrix and m n m n 2 2 and O 0 0 matrices. 0 0 0 0 and O 0 0 0 0 Remember that matrices are denoted by capital letters. So, when you solve for X, you are solving for a matrix that makes the matrix equation true. 2 3 zero matrix 2 2 zero |
matrix The algebra of real numbers and the algebra of matrices have many similarities. For example, compare the following solutions. Real Numbers (Solve for x.) Matrices (Solve for X.) The algebra of real numbers and the algebra of matrices also have important differences, which will be discussed later. Example 6 Solving a Matrix Equation Solve for X A 1 0 in the equation and 2 3 3X A B, B 3 2 where. 4 1 Solution Begin by solving the equation for X to obtain 3X B A X 1 3 B A. Now, using the matrices 3 2 A and, you have 2 3 Substitute the matrices. Subtract matrix A from matrix B. Multiply the matrix by 1 3. Now try Exercise 25. 333202_0802.qxd 12/5/05 10:57 AM Page 592 592 Chapter 8 Matrices and Determinants Matrix Multiplication The third basic matrix operation is matrix multiplication. At first glance, the definition may seem unusual. You will see later, however, that this definition of the product of two matrices has many practical applications. Definition of Matrix Multiplication B bij If AB matrix and m n matrix is an A aij m p is an AB cij is an n p matrix, the product where ci j ai1b1j ai2b2 j ai3b3j... ainbnj. The definition of matrix multiplication indicates a row-by-column multiplication, where the entry in the th row and is i by the corresponding obtained by multiplying the entries in the th row of B entries in the th column of and then adding the results. The general pattern for matrix multiplication is as follows. j th column of the product AB A j i b11 b21 b31... bn1 b12 b22 b32... bn2............ b1j b2j b3j... bnj............ b1p b2p b3p... bnp c11 c21... ci1... cm1 c12 c22... ci2... cm2............ c1j c2j... cij... cmj............ c1p c2 |
p... cip... cmp a11 a21 a31... ai1... am1 a12 a22 a32... ai2... am2 a13 a23 a33... ai3... am3............... a1n a2n a3n... ain... amn ai1b1j ai2b2j ai3b3j... ainbnj cij Example 7 Finding the Product of Two Matrices AB First, note that the product equal to the number of rows of Moreover, the product A find the entries of the product, multiply each row of as follows. A is defined because the number of columns of 3 2. B. AB by each column of has order is To B AB 1 13 34 9 43 24 53 04 4 15 1 6 10 12 31 42 21 52 01 Now try Exercise 29. 333202_0802.qxd 12/5/05 10:57 AM Page 593 Exploration Use the following matrices to find AB, BA, What do your results tell you about matrix multiplication, commutativity, and associativity? ABC. ABC, and Section 8.2 Operations with Matrices 593 Be sure you understand that for the product of two matrices to be defined, the number of columns of the first matrix must equal the number of rows of the second matrix. That is, the middle two indices must be the same. The outside two indices give the order of the product, as shown below. A m n B n p AB m p Equal Order of AB Example 8 Finding the Product of Two Matrices Find the product AB where A 1 2 0 1 3 2 and B 2 1 1. 4 0 1 Solution Note that the order of has order 2 2. A is 2 3 and the order of B is 3 2. So, the product AB AB 1 2 0 1 3 2 2 12 01 31 22 11 21 1 1 4 0 1 14 00 31 24 10 21 5 3 7 6 Now try Exercise 31. Example 9 Patterns in Matrix Multiplication a 10 5 9 3 1 c. The product AB for the following matrices is not defined and Now try Exercise 33. 333202_0802.qxd 12/5/05 10:57 AM Page 594 594 Chapter 8 Matrices and Determinants Example 10 Patterns in Matrix Multiplication a Now try Exercise 45 |
. In Example 10, note that the two products are different. Even if BA are defined, matrix multiplication is not, in general, commutative. That is, for most matrices, This is one way in which the algebra of real numbers and the algebra of matrices differ. AB BA. and AB Properties of Matrix Multiplication Let and be matrices and let be a scalar. c C B,A, ABC ABC AB C AB AC A B)C AC BC cAB cAB AcB 1. 2. 3. 4. Associative Property of Multiplication Distributive Property Distributive Property Associative Property of Scalar Multiplication Definition of Identity Matrix The is called the identity matrix of order n and is denoted by n n matrix that consists of 1’s on its main diagonal and 0’s elsewhere 1 0 0... 0 In. 0 1 0... 0 0 0 1...... 1 Identity matrix Note that an identity matrix must be square. When the order is understood to be you can denote simply by I. n, In If A is an n n For example, InA A. matrix, the identity matrix has the property that AIn A and and AI A IA A 333202_0802.qxd 12/5/05 10:57 AM Page 595 Section 8.2 Operations with Matrices 595 Applications Matrix multiplication can be used to represent a system of linear equations. Note how the system a11x1 a21x1 a31x1 a12x2 a22x2 a32x2 a13x3 a23x3 a33x3 b1 b2 b3 can be written as the matrix equation X of the system, and and B are column matrices. AX B, where A is the coefficient matrix... B represents A The notation the augmented matrix formed B is adjoined to when matrix... X matrix The notation represents the reduced rowechelon form of the augmented matrix that yields the solution to the system. I A. a11 a21 a31 a12 a22 a32 A a13 a23 a33 x1 x2 x3 b1 b2 b3 X B Example 11 Solving a System of Linear Equations Consider the following system of linear equations. x1 x3 2x3 2x3 2x2 x2 3x2 4 4 2 2x1 a. Write this system as a matrix equation, b. Use Gauss- |
Jordan elimination on the augmented matrix AX B. A B to solve for the matrix X. Solution a. In matrix form, AX B, the system can be written as follows x1 x2 x3 4 4 2 b. The augmented matrix is formed by adjoining matrix B to matrix A......... Using Gauss-Jordan elimination, you can rewrite this equation as......... So, the solution of the system of linear equations is x3 and the solution of the matrix equation is 1, 1, x1 x2 2, and X x1 x2 x3 1 2 1. Now try Exercise 55. 333202_0802.qxd 12/5/05 10:57 AM Page 596 596 Chapter 8 Matrices and Determinants Example 12 Softball Team Expenses Two softball teams submit equipment lists to their sponsors. Women’s Team Men’s Team Bats Balls Gloves 12 45 15 15 38 17 Each bat costs $80, each ball costs $6, and each glove costs $60. Use matrices to find the total cost of equipment for each team. Solution The equipment lists E and the costs per item C can be written in matrix form as E 12 45 15 15 38 17 and C 80 6 60. The total cost of equipment for each team is given by the product CE 80 6 6012 45 15 15 38 17 8012 645 6015 8015 638 6017 2130 2448. So, the total cost of equipment for the women’s team is $2130 and the total cost of equipment for the men’s team is $2448. Notice that you cannot find the total cost using the product is not defined. That is, the number of E columns of EC (2 columns) does not equal the number of rows of because (1 row). EC C Now try Exercise 63. W RITING ABOUT MATHEMATICS Problem Posing Write a matrix multiplication application problem that uses the matrix A 20 17 42 30. 33 50 Exchange problems with another student in your class. Form the matrices that represent the problem, and solve the problem. Interpret your solution in the context of the problem. Check with the creator of the problem to see if you are correct. Discuss other ways to represent and/or approach the problem. 333202_0802.qxd 12/5/05 10:57 AM Page 597 Section 8.2 Operations with Matrices 597 8.2 Exercises VOCAB |
ULARY CHECK: In Exercises 1–4, fill in the blanks. 1. Two matrices are ________ if all of their corresponding entries are equal. 2. When performing matrix operations, real numbers are often referred to as ________. 3. A matrix consisting entirely of zeros is called a ________ matrix and is denoted by ________. 4. The n n matrix consisting of 1’s on its main diagonal and 0’s elsewhere is called the ________ matrix of order n. In Exercises 5 and 6, match the matrix property with the correct form. m n, and and are scalars. d c B,A, and are matrices of order C 5. (a) (b) (c) (d) (e) 6. (a) (b) (c) (d) 1A A A B C A B C c dA cA dA cdA cdA A B B A A O A cAB AcB AB C AB AC ABC ABC (i) Distributive Property (ii) Commutative Property of Matrix Addition (iii) Scalar Identity Property (iv) Associative Property of Matrix Addition (v) Associative Property of Scalar Multiplication (i) Distributive Property (ii) Additive Identity of Matrix Addition (iii) Associative Property of Multiplication (iv) Associative Property of Scalar Multiplication PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. y. x 1. 2 y In Exercises 1–4, find and 2 4 22 7 5 x 12 8 2. x 7 5 y 16 x 2 3 0 1 7 3. 4. 13 8 16 2x 6 3 0 4 13 2 1 7 4 13 2 8 2y 2 5 15 4 4 6 0 3 2x y 2 2x 1 15 3y 5 8 18 2 4 3x 0 3 8 11 A B, (b) A B, 6. 5. and (d) In Exercises 5–12, if possible, find (a) 3A 2B. (c, 3A 10. 7. 3 1 4 2 9. 10. 11. 12 10 In Exercises 13–18, evaluate the expression. 13. 5 3 14 10 14 11 2 8 6 7 1 333202_0802.qxd 12/ |
5/05 10:57 AM Page 598 598 Chapter 8 Matrices and Determinants 15. 16. 17. 18. 1 2 44 0 5 2 30 18 4 9 2 1 3 3 0 14 6 24 3 7 1 7 1 4 13 4 6 8 65 1 9 6 3 0 3 2 11 1 3 9 5 1 1 In Exercises 19–22, use the matrix capabilities of a graphing utility to evaluate the expression. Round your results to three decimal places, if necessary. 0 2 19. 2 5 3 4 1 7 55 14 22 3.211 12 6 1.004 0.055 1 2 20. 21. 22. 63 2 11 22 19 13 20 6 1.630 6.829 4.914 3.889 20 9 5 14 8 7 5.256 9.768 15 6 0 31 3.090 8.335 4.251 19 10 10 16 24 In Exercises 23–26, solve fo X in the equation, given A [2 1 3 1 0 4] 23. 25. X 3A 2B 2X 3A B and B [ 0 2 4 3 0 1]. 24. 26. 2X 2A B 2A 4B 2X In Exercises 27–34, if possible, find the result. AB and state the order of 27. 28. 29. 30. 31. 32 17, 13 10 12, 33. 34 11 16 0 4 4 0 B 6 2 1 6 In Exercises 35– 40, use the matrix capabilities of a graphing utility to find if possible. AB, 35. 36. 37. 38. 39. 40. 2 10 A 5 A 11 14 6 6 5 5 12 10 2 3 1 5,, 4 12 9 A 3 12 5 8 15 1 6 9 1 A 2 21 13 10 12 16 8 4 B 1 B 12 5 15 B 3, B 2 24 16 8 0 15 14 7 32 8 6 5 0.5 1.6 2 4 9 1 15 10 4 6 14 21 10 A 9 100 B 52 40 A 15 4 8, 18 75 10 50 38 250 85 35 18 12 22, 45 27 82 60 B 7 8 22 16 1 24 In Exercises 41– 46, if possible, find (a) (Note: A2 AA. AB, (b) BA, and (c) A2 41. 42. 43. 44. 45 46 In Exercises 47–50, evaluate the expression |
. Use the matrix capabilities of a graphing utility to verify your answer. 1 47 333202_0802.qxd 12/5/05 10:57 AM Page 599 36 48. 49. 0 4 50 In Exercises 51–58, (a) write the system of linear equations and (b) use Gauss-Jordan as a matrix equation, elimination on the augmented matrix to solve for the matrix [A B] AX B, 51. 53. 55. 56. 57. 58. 3x3 x3 5x3 3x3 4 3x2 6x1 x2 36 x1 2x1 X. 4 x1 x2 2x1 x2 0 2x1 x1 x1 x1 5x2 x2 2x2 x2 3x2 6x2 2x2 3x2 5x2 x2 2x2 x2 x1 x1 x1 3x1 x1 x3 2x3 x3 5x3 4x3 5x3 52. 54. 2x1 3x2 5 x1 4x2 10 4x1 13 9x2 x1 3x2 12 9 6 17 9 6 5 20 8 16 17 11 40 59. Manufacturing A corporation has three factories, each of which manufactures acoustic guitars and electric guitars. The number of units of guitars produced at factory in one aij day is represented by. A 70 35 in the matrix 50 100 25 70 j Find the production levels if production is increased by 20%. 60. Manufacturing A corporation has four factories, each of which manufactures sport utility vehicles and pickup trucks. The number of units of vehicle produced at factory j in one day is represented by. A 100 40 in the matrix 90 20 30 60 70 60 aij i Find the production levels if production is increased by 10%. Section 8.2 Operations with Matrices 599 61. Agriculture A fruit grower raises two crops, apples and peaches. Each of these crops is sent to three different outlets for sale. These outlets are The Farmer’s Market, The Fruit Stand, and The Fruit Farm. The numbers of bushels of apples sent to the three outlets are 125, 100, and 75, respectively. The numbers of bushels of peaches sent to the three outlets are 100, 175, and 125, respectively. The profit per bushel for apples is $3.50 and the profit per bushel for peaches is $6.00. (a) |
Write a matrix i of each crop what each entry (b) Write a matrix A that represents the number of bushels that are shipped to each outlet State aij B that represents the profit per bushel of of the matrix of the matrix represents. j. bij each fruit. State what each entry represents. (c) Find the product matrix represents. BA and state what each entry of the 62. Revenue A manufacturer of electronics produces three models of portable CD players, which are shipped to two warehouses. The number of units of model that are j in the matrix shipped to warehouse is represented by i aij A 5,000 6,000 8,000. 4,000 10,000 5,000 The prices per unit are represented by the matrix B $39.50 $44.50 $56.50. Compute BA and interpret the result. 63. Inventory A company sells five models of computers through three retail outlets. The inventories are represented by S. Model Outlet The wholesale and retail prices are represented by T. Price Wholesale Retail T $840 $1200 $1450 $2650 $3050 $1100 $1350 $1650 $3000 $3200 A B C D E Model Compute ST and interpret the result. 333202_0802.qxd 12/5/05 10:57 AM Page 600 600 Chapter 8 Matrices and Determinants 64. Voting Preferences The matrix From R P 0.6 0.2 0.2 D 0.1 0.7 0.2 I 0.1 0.1 0.8 R D I To i j is called a stochastic matrix. Each entry represents the proportion of the voting population that changes from party represents the proportion that remains loyal to the party from one election to the next. Compute and interpret to party and P2. pij pii j, i 65. Voting Preferences Use a graphing utility to find P6, P7, P5, you detect a pattern as and P8 P 4, for the matrix given in Exercise 64. Can is raised to higher powers? P3, P 66. Labor/Wage Requirements A company that manufactures boats has the following labor-hour and wage requirements. Labor per boat Department Selling price Profit B 2.65 2.85 3.05 0.65 0.70 0.85 Skim milk 2% milk Whole milk (a) Compute AB and interpret the result. (b) Find the dairy mart’s total |
profit from milk sales for the weekend. 68. Profit At a convenience store, the numbers of gallons of 87-octane, 89-octane, and 93-octane gasoline sold over the A. weekend are represented by Octane 87 89 A 580 560 860 840 420 1020 93 320 160 540 Friday Saturday Sunday The selling prices per gallon and the profits per gallon for the three grades of gasoline sold by the convenience store are represents by B. Cutting Assembly Packaging S 1.0 hr 1.6 hr 2.5 hr 0.5 hr 1.0 hr 2.0 hr 0.2 hr 0.2 hr 1.4 hr Small Medium Large Boat size Selling price Profit B 1.95 2.05 2.15 0.32 0.36 0.40 87 89 93 Octane Wages per hour Plant A B T $12 $9 $8 $10 $8 $7 Cutting Assembly Packaging Department Compute ST and interpret the result. 67. Profit At a local dairy mart, the numbers of gallons of skim milk, 2% milk, and whole milk sold over the weekend are represented by A. Skim milk 2% Whole milk milk A 40 60 76 64 82 96 52 76 84 Friday Saturday Sunday The selling prices (in dollars per gallon) and the profits (in dollars per gallon) for the three types of milk sold by the dairy mart are represented by B. (a) Compute AB and interpret the result. (b) Find the convenience store’s profit from gasoline sales for the weekend. 69. Exercise The numbers of calories burned by individuals of different body weights performing different types of aerobic exercises for a 20-minute time period are shown in matrix A. Calories burned 120-lb person 150-lb person A 109 127 64 136 159 79 Bicycling Jogging Walking (a) A 120-pound person and a 150-pound person bicycled for 40 minutes, jogged for 10 minutes, and walked for 60 minutes. Organize the time spent exercising in a matrix B. (b) Compute BA and interpret the result. 333202_0802.qxd 12/5/05 10:57 AM Page 601 Model It 70. Health Care The health care plans offered this year by a local manufacturing plant are as follows. For individuals, the comprehensive plan costs $694.32, the HMO standard plan costs $451.80, and the HMO Plus plan costs $489.48. For families, the comprehensive plan costs $1725 |
.36, the HMO standard plan costs $1187.76 and the HMO Plus plan costs $1248.12. The plant expects the costs of the plans to change next year as follows. For individuals, the costs for the comprehensive, HMO standard, and HMO Plus plans will be $683.91, $463.10, and $499.27, respectively. For families, the costs for the comprehensive, HMO standard, and HMO Plus plans will be $1699.48, $1217.45, and $1273.08, respectively. A (a) Organize the information using two matrices and B, represents the health care plan costs for where this year and represents the health care plan costs for next year. State what each entry of each matrix represents. B A (b) Compute A B and interpret the result. (c) The employees receive monthly paychecks from which the health care plan costs are deducted. Use the matrices from part (a) to write matrices that show how much will be deducted from each employees’ paycheck this year and next year. (d) Suppose the costs of each plan instead increase by 4% next year. Write a matrix that shows the new monthly payment. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 71 and 72, determine whether 71. Two matrices can be added only if they have the same order. 6 2 72 In Exercises 73– 80, let matrices 3 2, Think About It and respectively. be of orders D Determine whether the matrices are of proper order to perform the operation(s). If so, give the order of the answer. C,B,A, 2 3, 2 3, 2 2, and 73. 75. 77. 79. A 2C AB BC D DA 3B 74. 76. 78. 80. B 3C BC CB D BC DA Section 8.2 Operations with Matrices 601 81. Think About It and a, b, then c If and are real numbers such that a b. c 0 ac bc, C and is not are nonzero matrices such that necessarily equal to Illustrate this using the following matrices. A 0 0 However, if AC BC, C 2 2 B 1 1 B,A, A then,, 1 1 3 3 0 0 B. b and b 0. a or AB O, are real numbers such that are |
it is not necessarily true that Illustrate this using the following However, if and A B 82. Think About It then If ab 0, a 0 matrices such that A O B O. or matrices 83. Exploration Let and be unequal diagonal matrices of the same order. (A diagonal matrix is a square matrix in which each entry not on the main diagonal is zero.) Determine the products for several pairs of such matrices. Make a conjecture about a quick rule for such products. AB 84. Exploration Let and let i. 0 and (a) Find i 3, A2, and i 4. A3, and A4. Identify any similarities with i 2, (b) Find and identify B2. Skills Review In Exercises 85–90, solve the equation. 85. 86. 87. 88. 89. 90. 3x2 20x 32 0 8x2 10x 3 0 4x3 10x2 3x 0 3x 3 22x2 45x 0 3x3 12x2 5x 20 0 2x3 5x2 12x 30 0 In Exercises 91–94, solve the system of linear equations both graphically and algebraically. 92. 91. x 4y 9 5x 8y 39 8x 3y 17 6x 7y 27 x 2y 5 8 3x y 94. 6x 13y 11 9x 5y 41 93. 333202_0803.qxd 12/5/05 11:01 AM Page 602 602 Chapter 8 Matrices and Determinants 8.3 The Inverse of a Square Matrix What you should learn • Verify that two matrices are inverses of each other. • Use Gauss-Jordan elimination to find the inverses of matrices. • Use a formula to find the inverses of 2 2 • Use inverse matrices to solve systems of linear equations. matrices. Why you should learn it You can use inverse matrices to model and solve real-life problems. For instance, in Exercise 72 on page 610, an inverse matrix is used to find a linear model for the number of licensed drivers in the United States. Jon Love/Getty Images The Inverse of a Matrix This section further develops the algebra of matrices. To begin, consider the real x, To solve this equation for multiply each side of the number equation equation by ax b. (provided that a 0 ). a1 ax b a1ax a1b |
1x a1b x a1b a1 The number definition of the multiplicative inverse of a matrix is similar. is called the multiplicative inverse of a because a1a 1. The A matrix and let be the Definition of the Inverse of a Square Matrix Let be an exists a matrix AA1 In A1 n n A1 A1A is called the inverse of The symbol such that n n then A1 A. In identity matrix. If there is read “ A inverse.” Example 1 The Inverse of a Matrix Show that is the inverse of where A, B A 1 1 2 1 and B 1 1. 2 1 Solution To show that B AB 1 1 BA 1 1 2 1 is the inverse of 1 2 1 1 1 1 A, show that AB I BA as follows. 0 1 0 1 As you can see, an inverse. Note that not all square matrices have an inverse. AB I BA. This is an example of a square matrix that has Now try Exercise 1. Recall that it is not always true that A So, in Example 1, you need only to check that AB I2. even if both products are AB In, it can be defined. However, if BA In. shown that are both square matrices and and B AB BA, 333202_0803.qxd 12/5/05 11:01 AM Page 603 Section 8.3 The Inverse of a Square Matrix 603 Finding Inverse Matrices A A A and is of order BA is called invertible (or nonsingular); otherwise, If a matrix has an inverse, A is called singular. A nonsquare matrix cannot have an inverse. To see this, note ), the products that if AB are of different orders and so cannot be equal to each other. Not all square matrices have inverses (see the matrix at the bottom of page 605). If, however, a matrix does have an inverse, that inverse is unique. Example 2 shows how to use a system of equations to find the inverse of a matrix. is of order m n (where n m m n and B Example 2 Finding the Inverse of a Matrix Find the inverse of A 1 1 4 3. Solution To find the inverse of A, try to solve the matrix equation AX I for X. X I A 1 1 4x21 3x21 x11 x11 4 3 x11 x12 x22 x21 4x22 3x22 x12 x |
12 1 0 1 0 0 1 0 1 Equating corresponding entries, you obtain two systems of linear equations. x11 x11 x12 x12 4x21 3x21 4x22 3x22 1 0 0 1 Linear system with two variables, x11 and x21. Linear system with two variables, x12 and x22. Solve the first system using elementary row operations to determine that 4 x11 and x21 Therefore, the inverse of From the second system you can determine that 1. and x12 is A 3 1. x22 X A1 3 1. 4 1 You can use matrix multiplication to check this result. Check AA1 1 1 A1A ✓ ✓ Now try Exercise 13. 333202_0803.qxd 12/5/05 11:01 AM Page 604 604 Chapter 8 Matrices and Determinants In Example 2, note that the two systems of linear equations have the same coefficient matrix 1 1 4 3 A....... and 1 1 4 3...... Rather than solve the two systems represented by 1 0 0 1 separately, you can solve them simultaneously by adjoining the identity matrix to the coefficient matrix to obtain I A 1 1 4 3...... 1 0. 0 1 This “doubly augmented” matrix can be represented as By applying Gauss-Jordan elimination to this matrix, you can solve both systems with a single elimination process. A I. Te c h n o l o g y Most graphing utilities can find the inverse of a square matrix. To do so, you may have to use the inverse key. Consult the user’s guide for your graphing utility for specific keystrokes.................. 1 0 1 1 3 1 R1 4R2 R2 R1 0 1 0 1 4 1 A I, So, from the “doubly augmented” matrix I A1. you obtain the matrix A 1 1 4 3...... A1 3 1 4 1...... This procedure (or algorithm) works for any square matrix that has an inverse. Finding an Inverse Matrix Let be a square matrix of order A n. 1. Write the and the n 2n matrix that consists of the given matrix on the left n n identity matrix on the right to obtain I A A I. 2. If possible, row reduce A I. A entire matrix not possible, is not invertible. I |
to using elementary row operations on the A The result will be the matrix I A1. If this is 3. Check your work by multiplying to see that AA1 I A1A. 333202_0803.qxd 12/8/05 10:44 AM Page 605 Section 8.3 The Inverse of a Square Matrix 605 Example 3 Finding the Inverse of a Matrix Find the inverse of Solution Begin by adjoining the identity matrix to 1 0 2............ A to form the matrix 0 1 0. 0 0 1 I A1, as follows. Use elementary row operations to obtain the form R1 6R1 R2 R2 R3 R1 4R2 R3 R3 R3 R1 R2 → 1 1 →.............................. A1 Be sure to check your solution because it is easy to make algebraic errors when using elementary row operations. So, the matrix A is invertible and its inverse is A1 Confirm this result by multiplying A and A1 to obtain I, as follows. Check AA1 Now try Exercise 21. The process shown in Example 3 applies to any A matrix When using this algorithm, if the matrix does not reduce to the identity matrix, then does not have an inverse. For instance, the following matrix has no inverse. n n A above has no inverse, adjoin the identity matrix to to To confirm that matrix form and perform elementary row operations on the matrix. After doing so, you will see that it is impossible to obtain the identity matrix on the left. Therefore, is not invertible. A I A 333202_0803.qxd 12/5/05 11:01 AM Page 606 606 Chapter 8 Matrices and Determinants Exploration Use a graphing utility with matrix capabilities to find the inverse of the matrix 3. A 1 2 6 What message appears on the screen? Why does the graphing utility display this message? The Inverse of a 2 2 Matrix Using Gauss-Jordan elimination to find the inverse of a matrix works well (even 2 2 as a computer technique) for matrices of order matrices, however, many people prefer to use a formula for the inverse rather 2 2 than Gauss-Jordan elimination. This simple formula, which works only for A matrices, is explained as follows. If or greater. For matrix given by 3 3 2 2 is a A a |
c b d is invertible if and only if A then inverse is given by ad bc 0. Moreover, if ad bc 0, the A1 1 ad bc d c ad bc The denominator will study determinants in the next section. b a. Formula for inverse of matrix A is called the determinant of the 2 2 matrix You A. Example 4 Finding the Inverse of a 2 2 Matrix If possible, find the inverse of each matrix. a. Solution a. For the matrix A, apply the formula for the inverse of a 2 2 matrix to obtain ad bc 32 12 4. Because this quantity is not zero, the inverse is formed by interchanging the entries on the main diagonal, changing the signs of the other two entries, and multiplying by the scalar as follows. 1 4, A1 1 2 42 Substitute for a, b, c, d, and the determinant. Multiply by the scalar 1 4. b. For the matrix B, you have ad bc 32 16 0 which means that B is not invertible. Now try Exercise 39. 333202_0803.qxd 12/5/05 11:01 AM Page 607 Section 8.3 The Inverse of a Square Matrix 607 Systems of Linear Equations You know that a system of linear equations can have exactly one solution, of a square infinitely many solutions, or no solution. If the coefficient matrix system (a system that has the same number of equations as variables) is invertible, the system has a unique solution, which is defined as follows. A A System of Equations with a Unique Solution A If AX B has a unique solution given by is an invertible matrix, the system of linear equations represented by Te c h n o l o g y X A1B. To solve a system of equations with a graphing utility, enter the B matrices and editor. Then, using the inverse key, solve for in the matrix X. A A x 1 B ENTER The screen will display the solution, matrix X. Example 5 Solving a System Using an Inverse You are going to invest $10,000 in AAA-rated bonds, AA-rated bonds, and B-rated bonds and want an annual return of $730. The average yields are 6% on AAA bonds, 7.5% on AA bonds, and 9.5% on B bonds. You will invest twice as much in AAA bonds as in B bonds. Your investment can be represented |
as x 0.06x x y 0.075y z 0.095z 2z 10,000 730 0 x, where respectively. Use an inverse matrix to solve the system. represent the amounts invested in AAA, AA, and B bonds, and y, z Solution Begin by writing the system in the matrix form AX B. 1 0.06 1 1 0.075 0 1 0.095 2 x y z 10,000 730 0 Then, use Gauss-Jordan elimination to find A1. A1 15 21.5 7.5 Finally, multiply by B X A1B 200 300 100 A1 2 3.5 1.5 on the left to obtain the solution. 15 21.5 7.5 200 300 100 2 3.5 1.5 10,000 4000 4000 2000 730 0 y 4000, x 4000, The solution to the system is invest $4000 in AAA bonds, $4000 in AA bonds, and $2000 in B bonds. z 2000. and So, you will Now try Exercise 67. 333202_0803.qxd 12/5/05 11:01 AM Page 608 608 Chapter 8 Matrices and Determinants 8.3 Exercises VOCABULARY CHECK: Fill in the blanks. 1. In a ________ matrix, the number of rows equals the number of columns. 2. If there exists an n n matrix A1 such that AA1 In A1 A, then A1 is called the ________ of A. 3. If a matrix has an inverse, it is called invertible or ________; if it does not have an inverse, A it is called ________. 4. If A is an invertible matrix, the system of linear equations represented by AX B has a unique solution given by X ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. 1. 3. 2, In Exercises 1–10, show that B is the inverse of A 17 11 3 2 3 5 11. 5. 6 11 4 7 4 2 9 1 5 3 14 10, 34, 12 33. 8. 9. 10. 4 8 17. 16. 14. 13. 18. 11. 15. 12. 2 7 1 1 0 3 2 3 33 19 1 0 3 4 In Exercises 11–26, find the inverse of the matrix (if |
it exists). 11 15 20. 23. 25. 26. 24. 22. 19. 21. 3 2 6 0 3 3 3 2 In Exercises 27–38, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 27. 29. 31 10 15 2 0 3 1 4 3 2 1 2 28. 30. 5 3 10 3 32 11 6 2 5 2 333202_0803.qxd 12/5/05 11:01 AM Page 609 Section 8.3 The Inverse of a Square Matrix 609 33. 35. 37. 0.3 0.5 0..2 0.2 0.4 0.3 0.2 0. 34. 36. 38. 0.7 1 0..3 0.2 0.9 14 6 7 10 2 3 5 11 1 2 2 4 39. 2 2 In Exercises 39–44, use the formula on page 606 to find the inverse of the matrix (if it exists). 2 5 7 8 3 2 6 4 12 3 2 5 7 1 12 5 44. 40. 43. 42. 41 In Exercises 45– 48, use the inverse matrix found in Exercise 13 to solve the system of linear equations. 45. 47. x 2y 5 2x 3y 10 x 2y 4 2x 3y 2 46. 48. x 2y 0 2x 3y 3 x 2y 2x 3y 1 2 In Exercises 49 and 50, use the inverse matrix found in Exercise 21 to solve the system of linear equations. 49. x y z 0 3x 5y 4z 5 3x 6y 5z 2 50. x y z 3x 5y 4z 3x 6y 5z 1 2 0 51. 3x1 2x1 x1 In Exercises 51 and 52, use the inverse matrix found in Exercise 38 to solve the system of linear equations. 2x4 3x4 5x4 11x4 2x4 3x4 5x4 11x4 x3 2x3 2x3 4x3 x3 2x3 2x3 4x3 2x2 5x2 5x2 4x2 2x2 5x2 5x2 4x2 x1 x1 1 2 0 3 0 1 1 2 3x1 2x1 x1 52. In Exercises 53– 60, use an inverse matrix to solve (if |
possible) the system of linear equations. 53. 3x 4y 5x 3y 2 4 54. 18x 12y 13 30x 24y 23 55. 57. 59. 1.6 0.8y 4y 5 8 y 2 4 y 12 0.4x 2x 1 4 x 3 2 x 3 4x y z 2x 2y 3z 5x 2y 6z 3 5 10 1 56. 58. 60. 4 0.6y 1.4y y 20 2 y 51 0.2x x 5 6 x 3 x 4x 2y 3z 2x 2y 5z 8x 5y 2z 7 2.4 8.8 2 16 4 In Exercises 61–66, use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. 61. 63. 64. 65. 66. 62. 2x 3y 5z 4 3x 5y 9z 7 5x 9y 17z 13 29 37 24 151 86 187 2 3 4 2x 2y 3z x 7y 8z 4x y 3z x 5y z 5x 3y 2z 3x 2y z 8x 7y 10z 7x 2x 12x 3y 5z 15x 9y 2z 3y y 2w w z 2w w w 2z 2w 5z w 3w 5y 4y 2y 2x 4x x x 2x x y 41 13 12 8 11 7 3 1 Investment Portfolio In Exercises 67–70, consider a person who invests in AAA-rated bonds, A-rated bonds, and B-rated bonds. The average yields are 6.5% on AAA bonds, 7% on A bonds, and 9% on B bonds. The person invests z twice as much in B bonds as in A bonds. Let represent the amounts invested in AAA, A, and B bonds, respectively. and x, y, 0.065x x y 0.07y 2y z 0.09z z (total investment) (annual return) 0 Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond. Total Investment Annual Return 67. $10,000 68. $10,000 69. $12,000 70. $500,000 $705 $760 $835 $38,000 333202_0803.qxd 12/5/05 11: |
01 AM Page 610 610 Chapter 8 Matrices and Determinants 71. Circuit Analysis Consider the circuit shown in the figure. in amperes, are the solution of I2, I3, and I1, The currents the system of linear equations 4I3 4I3 I3 2I1 I2 I2 E1 E2 0 I1 E2 E1 and where are voltages. Use the inverse of the coefficient matrix of this system to find the unknown currents for the voltages. 2Ω d + _ I1 E1 I3 I2 4Ω 1Ω b + _ E2 a c (a) (b) E1 E1 14 volts, E2 24 volts, E2 28 volts 23 volts Model It 72. Data Analysis: Licensed Drivers The table shows the numbers (in millions) of licensed drivers in the United States for selected years 1997 to 2001. (Source: U.S. Federal Highway Administration) y Year Drivers, y 1997 1999 2001 182.7 187.2 191.3 (a) Use the technique demonstrated in Exercises 57–62 in Section 7.2 to create a system of linear equations for the data. Let represent the year, with t 7 corresponding to 1997. t (b) Use the matrix capabilities of a graphing utility to find an inverse matrix to solve the system from part (a) and find the least squares regression line y at b. (c) Use the result of part (b) to estimate the number of licensed drivers in 2003. (d) The actual number of licensed drivers in 2003 was 196.2 million. How does this value compare with your estimate from part (c)? Model It (co n t i n u e d ) (e) Use the result of part (b) to estimate when the number of licensed drivers will reach 208 million. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 73 and 74, determine whether 73. Multiplication of an invertible matrix and its inverse is commutative. 74. If you multiply two square matrices and obtain the identity matrix, you can assume that the matrices are inverses of one another. 75. If A is a 2 2 matrix if and only if inverse is ad bc 0., A a b d c ad bc 0, then If A is invertible verify that the A1 1 ad bc d c. b a 76 |
. Exploration Consider matrices of the form A a11 0 0 0 0 a22 0 0 0 0 a33 ann (a) Write a A. matrix and a Find the inverse of each. 2 2 3 3 matrix in the form of (b) Use the result of part (a) to make a conjecture about the inverses of matrices in the form of A. Skills Review In Exercises 77 and 78, solve the inequality and sketch the solution on the real number line. 77. x 7 ≥ 2 78. 2x 1 < 3 In Exercises 79– 82, solve the equation. Approximate the result to three decimal places. 79. 81. 3x2 315 log2 x 2 4.5 80. 82. 2000ex5 400 ln x lnx 1 0 83. Make a Decision To work an extended application analyzing the number of U.S. households with color televisions from 1985 to 2005, visit this text’s website at college.hmco.com. Source: Nielsen Media Research) (Data 333202_0804.qxd 12/5/05 11:03 AM Page 611 Section 8.4 The Determinant of a Square Matrix 611 8.4 The Determinant of a Square Matrix What you should learn • Find the determinants of 2 2 matrices. • Find minors and cofactors of square matrices. • Find the determinants of square matrices. Why you should learn it Determinants are often used in other branches of mathematics. For instance, Exercises 79–84 on page 618 show some types of determinants that are useful when changes in variables are made in calculus. The Determinant of a 2 2 Matrix Every square matrix can be associated with a real number called its determinant. Determinants have many uses, and several will be discussed in this and the next section. Historically, the use of determinants arose from special number patterns that occur when systems of linear equations are solved. For instance, the system a1x b1y c1 a2x b2y c2 has a solution x c1b2 a1b2 c2b1 a2b1 a1b2 and y a1c2 a1b2 a 2c1 a 2b1 provided that Note that the denominators of the two fractions are the same. This denominator is called the determinant of the coefficient matrix of the system. a2b1 |
0. Coefficient Matrix A a1 a 2 b1 b2 Determinant detA a1b2 a 2b1 The determinant of the matrix can also be denoted by vertical bars on both sides of the matrix, as indicated in the following definition. A Definition of the Determinant of a 2 The determinant of the matrix 2 Matrix A a1 a2 b1 b2 is given by detA A a1 a 2 b1 b2 a1b2 a 2b1. detA A In this text, A. are used interchangeably to represent the determinant of Although vertical bars are also used to denote the absolute value of a real number, the context will show which use is intended. and A convenient method for remembering the formula for the determinant of a 2 2 matrix is shown in the following diagram. detA a1 a 2 b1 b2 a1b2 a 2b1 Note that the determinant is the difference of the products of the two diagonals of the matrix. 333202_0804.qxd 12/5/05 11:03 AM Page 612 612 Chapter 8 Matrices and Determinants Example 1 The Determinant of a 2 2 Matrix Find the determinant of each matrix. a. b. c detA 2 2 1 Solution 3 2 Exploration Use a graphing utility with matrix capabilities to find the determinant of the following matrix. A 1 1 3 2 0 2 What message appears on the screen? Why does the graphing utility display this message? a. b. c. 22 13 4 3 7 detB 2 4 detC 0 2 1 2 3 2 4 22 41 4 4 0 04 23 0 3 3 2 Now try Exercise 5. Notice in Example 1 that the determinant of a matrix can be positive, zero, or negative. The determinant of a matrix of order 1 1 is defined simply as the entry of the matrix. For instance, if A 2, then detA 2. Te c h n o l o g y Most graphing utilities can evaluate the determinant of a matrix. For instance, you can evaluate the determinant of A 2 1 3 2 by entering the matrix as The result should be 7, as in Example 1(a). Try evaluating the determinants of other matrices. Consult the user’s guide for your graphing utility for specific keystrokes. and then choosing the determinant feature. A 333202_0804.qxd 12/ |
5/05 11:03 AM Page 613 Section 8.4 The Determinant of a Square Matrix 613 Minors and Cofactors To define the determinant of a square matrix of order convenient to introduce the concepts of minors and cofactors. 3 3 or higher, it is Sign Pattern for Cofactors...... 3 3 matrix 4 4... n n matrix... matrix.................. A is a square matrix, the minor Minors and Cofactors of a Square Matrix Mi j ai j of the entry If is the determinant i j of the matrix obtained by deleting the th row and th column of The cofactor of the entry A. is Ci j ai j Ci j 1ijMi j. In the sign pattern for cofactors at the left, notice that odd positions (where is even) have is odd) have negative signs and even positions (where i j i j positive signs. Example 2 Finding the Minors and Cofactors of a Matrix Find all the minors and cofactors of Solution To find the minor determinant of the resulting matrix. M11, delete the first row and first column of A and evaluate the M11 2 1 11 02 1 Similarly, to find M12, delete the first row and second column M12 2 1 31 42 5 Continuing this pattern, you obtain the minors. M11 M21 M31 1 2 5 M12 M22 M32 5 4 3 M13 M23 M33 4 8 6 3 3 matrix shown at the upper left. Now, to find the cofactors, combine these minors with the checkerboard pattern of signs for a 1 2 5 4 8 6 5 4 C13 C11 C22 C12 C23 C21 3 C33 C32 C31 Now try Exercise 27. 333202_0804.qxd 12/5/05 11:03 AM Page 614 614 Chapter 8 Matrices and Determinants The Determinant of a Square Matrix The definition below is called inductive because it uses determinants of matrices of order to define determinants of matrices of order n 1 n. A is a square matrix (of order Determinant of a Square Matrix 2 2 If or greater), the determinant of the sum of the entries in any row (or column) of multiplied by their respective cofactors. For instance, expanding along the first row yields A |
A is A a11C11 a12C12... a1nC1n. Applying this definition to find a determinant is called expanding by cofactors. 2 2 matrix Try checking that for a A a1 a2 b1 b2 this definition of the determinant yields defined. A a1b2 a2b1, as previously Example 3 The Determinant of a Matrix of Order 3 3 Find the determinant of Solution Note that this is the same matrix that was in Example 2. There you found the cofactors of the entries in the first row to be 4. 1, 5, and C13 C11 C12 So, by the definition of a determinant, you have a12C12 A a11C11 a13C13 First-row expansion 01 25 14 14. Now try Exercise 37. In Example 3, the determinant was found by expanding by the cofactors in the first row. You could have used any row or column. For instance, you could have expanded along the second row to obtain a 23C23 a 22C22 32 14 28 14. A a 21C21 Second-row expansion 333202_0804.qxd 12/5/05 11:03 AM Page 615 Section 8.4 The Determinant of a Square Matrix 615 When expanding by cofactors, you do not need to find cofactors of zero entries, because zero times its cofactor is zero. aijCij 0Cij 0 So, the row (or column) containing the most zeros is usually the best choice for expansion by cofactors. This is demonstrated in the next example. Example 4 The Determinant of a Matrix of Order 4 4 Find the determinant of Solution After inspecting this matrix, you can see that three of the entries in the third column are zeros. So, you can eliminate some of the work in the expansion by using the third column. A 3C13 C23, C33, To do this, delete the first row and third column of 0C33 have zero coefficients, you need only find the cofacand evaluate the 0C23 C43 and Because C13. tor determinant of the resulting matrix. 0C43 A 0 3 1131 1 0131 1 2 4 0 3 4 C13 C13 2141 2 3 Expanding by cofactors in the second row yields Delete 1st row and 3rd column. Simplify. 2 2 3 |
151 3 1 4 0 218 317 5. So, you obtain A 3C13 35 15. Now try Exercise 47. Try using a graphing utility to confirm the result of Example 4. 333202_0804.qxd 12/5/05 11:03 AM Page 616 616 Chapter 8 Matrices and Determinants 8.4 Exercises VOCABULARY CHECK: Fill in the blanks. A 1. Both Mij 2. The ________ detA and aij column of the square matrix of the entry A. represent the ________ of the matrix A. is the determinant of the matrix obtained by deleting the th row and th j i 3. The ________ Cij of the entry aij of the square matrix A is given by 1ij Mij. 4. The method of finding the determinant of a matrix of order 2 2 or greater is called ________ by ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–16, find the determinant of the matrix. 1. 3. 5. 7. 9. 11. 13. 15. 4. 6. 8. 10. 12. 14. 16 In Exercises 17–22, use the matrix capabilities of a graphing utility to find the determinant of the matrix. 17. 19. 21. 0.2 0.4 0.3 0.9 1 0.1 2.2 3 2 0.2 0.2 0.4 0.7 0.3 4.2 4 6 1 0.2 0.2 0.3 0 1.3 6.1 2 6 4 18. 20. 22. 0.3 0.5 0.1 0.1 2 7.5 0.3 0 0 0.2 0.2 0.4 0.3 0.2 0.4 4.3 0.7 1.2 1 2 2 0.1 6.2 0.6 3 5 0 In Exercises 23–30, find all (a) minors and (b) cofactors of the matrix. 3 2 3 2 11 3 6 7 5 2 1 4 4 5 23. 25. 26. 24. 0 2 27. 29 28. 30 In Exercises 31–36, find the determinant of the matrix by the method of expansion by cofactors. Expand using the |
indicated row or column. 31. 33. 3 2 5 3 (a) Row 1 4 2 (b) Column 2 3 4 3 0 12 6 5 0 1 1 6 1 32. 34. 3 6 4 4 3 7 (a) Row 2 2 1 8 (b) Column 3 5 0 10 10 30 0 5 10 1 (a) Row 2 (b) Column 2 (a) Row 3 (b) Column 1 35. 6 4 1 8 0 13 36. 10 a) Row 2 (b) Column 2 (a) Row 3 (b) Column 1 In Exercises 37–52, find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. 37 38 333202_0804.qxd 12/5/05 11:03 AM Page 617 Section 8.4 The Determinant of a Square Matrix 617 39. 41. 43. 45. 47. 49. 51. 52. 53. 55. 57. 59. 60 12 11 2 6 0 1 3 4 6 0 2 40. 42. 44. 46. 48. 50. 5 2 0 0 0 In Exercises 53– 60, use the matrix capabilities of a graphing utility to evaluate the determinant 12 14 14 4 12 54. 56. 58 In Exercises 61– 68, (d) find (a) A, (b) B, (c) AB, and AB 61. 62. 63. 64. 65. 66. 67. 68 In Exercises 69 –74, evaluate the determinant(s) to verify the equation. cx z x y y x x w x cw x z y z cz cw z cy z w cx 0 z 2 y xz xz y a b b23a cw 71. 69. 73. 70. 72 78. x 4 3 74. 75. 76. 77. a a 1 3 7 In Exercises 75–78, solve for x 333202_0804.qxd 12/8/05 10:39 AM Page 618 618 Chapter 8 Matrices and Determinants In Exercises 79–84, evaluate the determinant in which the entries are functions. Determinants of this type occur when changes in variables are made in calculus. 92. If A of is obtained from A B by adding a multiple of a row of to another row of or by adding a multiple of a column |
A to another column of B A. then A, A 1 1 2v 4u 3e3x e2x 1x x 2e2x ln x e3x 1 79. 81. 83. 1 3x 2 ex x 1 ex 3y 2 1 1 xex 1 ln x x ln x xex 80. 82. 84. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 85 and 86, determine whether 85. If a square matrix has an entire row of zeros, the determi- nant will always be zero. 86. If two columns of a square matrix are the same, the determinant of the matrix will be zero. 87. Exploration Find square matrices and A B to demonstrate that A B A B. 88. Exploration Consider square matrices in which the entries are consecutive integers. An example of such a matrix is 4 7 10 5 8 11. 6 9 12 (a) Use a graphing utility to evaluate the determinants of four matrices of this type. Make a conjecture based on the results. (b) Verify your conjecture. 89. Writing Write a brief paragraph explaining the differ- ence between a square matrix and its determinant. 90. Think About It A 5, If is it possible to find is a matrix of order 2A? Explain. A 3 3 such that Properties of Determinants In Exercises 91–93, a property of determinants is given ( are square matrices). and State how the property has been applied to the given determinants and use a graphing utility to verify the results. A B 91. If B is obtained from by interchanging two rows of or A, interchanging two columns of B A. then A A 7 6 1 1 2 1 (a) (b is obtained from by multiplying a row by a nonzero or by multiplying a column by a nonzero then B cA. constant constant c c, A (a) (b) 93. If a) (b) (a) (c) 3 0 3 17 10 3 6 3 51 10 2 2 3 3 121 3 7 3 6 8 12 b 94. Exploration A diagonal matrix is a square matrix with all zero entries above and below its main diagonal. Evaluate the determinant of each diagonal matrix. Make a conjecture based on your results. Skills Review In Exercises 95–100, find the domain of the function. 95. f |
x x3 2x 96. gx 3x 97. hx 16 x2 99. gt lnt 1 98. 100. Ax 3 36 x2 f s 625e0.5s In Exercises 101 and 102, sketch the graph of the solution of the system of inequalities. 101. x x 2x y ≤ ≥ y < 8 3 5 102. x y > y ≤ 7x 4y ≤ 4 1 10 In Exercises 103–106, find the inverse of the matrix (if it exists). 103. 105 104. 5 3 106 333202_0805.qxd 12/5/05 11:05 AM Page 619 Section 8.5 Applications of Matrices and Determinants 619 8.5 Applications of Matrices and Determinants What you should learn • Use Cramer’s Rule to solve systems of linear equations. • Use determinants to find the areas of triangles. • Use a determinant to test for collinear points and find an equation of a line passing through two points. • Use matrices to encode and decode messages. Why you should learn it You can use Cramer’s Rule to solve real-life problems. For instance, in Exercise 58 on page 630, Cramer’s Rule is used to find a quadratic model for the number of U.S. Supreme Court cases waiting to be tried. Cramer’s Rule So far, you have studied three methods for solving a system of linear equations: substitution, elimination with equations, and elimination with matrices. In this section, you will study one more method, Cramer’s Rule, named after Gabriel Cramer (1704–1752). This rule uses determinants to write the solution of a system of linear equations. To see how Cramer’s Rule works, take another look at the solution described at the beginning of Section 8.4. There, it was pointed out that the system a1x b1y c1 a2x b2y c2 has a solution x c1b2 a1b2 c2b1 a2b1 a1b2 and y a1c2 a1b2 a2c1 a2b1 a2b1 provided that can be expressed as a determinant, as follows. 0. Each numerator and denominator in this solution x c1b2 a1b2 c2b1 a2b1 y a1 |
c2 a1b2 a2c1 a2b1 c2 c1 a1 a2 b1 b2 b2 b1 a2 a1 a1 a2 c1 c2 b2 b1 Relative to the original system, the denominator for and is simply the deterD. minant of the coefficient matrix of the system. This determinant is denoted by respectively. They are The numerators for formed by using the column of constants as replacements for the coefficients of x are denoted by as follows. and and and Dy, Dx y, x y y x Coefficient Matrix a1 b1 b2 a2 D a1 a2 b1 b2 Dx c1 c2 b1 b2 Dy a1 a2 c1 c2 © Lester Lefkowitz /Corbis For example, given the system 2x 5y 3 4x 3y 8 the coefficient matrix, D, Dx, and Dy are as follows. Coefficient Matrix Dx 5 3 Dy 2 4 3 8 333202_0805.qxd 12/5/05 11:05 AM Page 620 620 Chapter 8 Matrices and Determinants Cramer’s Rule generalizes easily to systems of equations in variables. The value of each variable is given as the quotient of two determinants. The denominator is the determinant of the coefficient matrix, and the numerator is the determinant of the matrix formed by replacing the column corresponding to the variable (being solved for) with the column representing the constants. For instance, the solution for in the following system is shown. n n x3 a11x1 a21x1 a31x1 a12x2 a22x2 a32x2 a13x3 a23x3 a33x3 b1 b2 b3 x3 A3 A a21 a31 a11 a11 a21 a31 a12 a22 a32 a12 a22 a32 b1 b2 b3 a33 a13 a23 n A, linear equations in variables has a coefficient matrix Cramer’s Rule n If a system of with a nonzero determinant A1 A2 A, A, i Ai where the th column of equations. If the determinant of the coefficient matrix is zero, the system has either no solution or infinitely many solutions. the solution of the system is An A is the column of constants in the system of..., x2 x1 xn |
A Example 1 Using Cramer’s Rule for a 2 2 System Use Cramer’s Rule to solve the system of linear equations. 4x 2y 10 3x 5y 11 Because this determinant is not zero, you can apply Cramer’s Rule. Solution To begin, find the determinant of the coefficient matrix. D 4 3 x Dx D y Dy D 2 5 20 6 14 10 4 50 22 14 5 11 14 2 11 10 3 44 30 14 y 1. and 14 x 2 Now try Exercise 1. 28 14 2 14 14 1 So, the solution is Check this in the original system. 333202_0805.qxd 12/5/05 11:05 AM Page 621 Section 8.5 Applications of Matrices and Determinants 621 Example 2 Using Cramer’s Rule for a 3 3 System Use Cramer’s Rule to solve the system of linear equations. x 2x 3x 2y 4y 3z 1 z 0 4z 2 Solution To find the determinant of the coefficient matrix 1 2 3 2 0 4 3 1 4 expand along the second row, as follows. D 213 2 3 4 0141 3 3 4 1151 3 4 2 4 24 0 12 10 Because this determinant is not zero, you can apply Cramer’s Rule 10 1 0 2 10 2 0 4 10 2, 8 5 x Dx D y Dy D z Dz D 8 10 4 5 15 10 3 2 16 10 8 5 The solution is 4 5, 3. Check this in the original system as follows. Check 4 5 4 5 24 5 8 5 34 5 12 5 23 2 3 8 5 8 5 43 2 6 38 5 24 5 48 5 32 Now try Exercise 7. Substitute into Equation 1. Equation 1 checks. ✓ Substitute into Equation 2. Equation 2 checks. ✓ Substitute into Equation 3. Equation 3 checks. ✓ Remember that Cramer’s Rule does not apply when the determinant of the coefficient matrix is zero. This would create division by zero, which is undefined. 333202_0805.qxd 12/5/05 11:05 AM Page 622 622 Chapter 8 Matrices and Determinants Area of a Triangle Another application of matrices and determinants is finding the area of a triangle whose vertices are given as points in a coordinate plane. Area of a Triangle The area of a triangle with vertices x |
1, y1, x2, y2, and x3, y3 is x1 1 2 y1 y2 y3 1 1 1 Area ± x2 x3 ± where the symbol yield a positive area. indicates that the appropriate sign should be chosen to Example 3 Finding the Area of a Triangle (4, 3) Find the area of a triangle whose vertices are in Figure 8.1. 1, 0, 2, 2, and 4, 3, as shown Solution x1, y1 Let of the triangle, evaluate the determinant. 1, 0, x2, y2 2, 2, and x3, y3 4, 3. Then, to find the area y 3 2 1 (1, 0) FIGURE 8.1 (2, 2) 1 2 3 4 x x1 x2 x3 y1 y2 y3 1 1 1142 4 2 3 1 1 1 1 0132 1 4 11 0 12 3 1122 Using this value, you can conclude that the area of the triangle is Area 1 2 Choose so that the area is positive. 1 2 3 3 2 square units. Now try Exercise 19. Exploration Use determinants to find the area of a triangle with vertices and and using the formula 7, 1, Confirm your answer by plotting the points in a coordinate plane 3, 1, 7, 5. Area 1 2 baseheight. 333202_0805.qxd 12/5/05 11:05 AM Page 623 y 3 2 1 (0, 1) FIGURE 8.2 (4, 3) (2, 21 (−2, −2) FIGURE 8.3 (7, 5) (1, 1) 1 2 3 4 5 6 7 x Section 8.5 Applications of Matrices and Determinants 623 Lines in a Plane What if the three points in Example 3 had been on the same line? What would have happened had the area formula been applied to three such points? The answer is that the determinant would have been zero. Consider, for instance, the 0, 1, 2, 2, three collinear points as shown in Figure 8.2. The area of the “triangle” that has these three points as vertices is 4, 3, and 20 2 4 1 1 2 3 1 1 20122 1 1 3 1 1 1132 4 1 1 1142 4 2 3 0 12 12 1 2 0. The result is generalized as follows. |
and x3, y3 are collinear (lie on the same line) Test for Collinear Points x1, y1 Three points if and only if, x2, y2, x1 x2 x3 y1 y2 y3 1 0. 1 1 Example 4 Testing for Collinear Points Determine whether the points Figure 8.3.) 2, 2, 1, 1, and 7, 5 are collinear. (See Solution Letting x1 x2 x3 x1, y1 y1 y2 y3 x3, y3 2, 2, x2, y2 1 1 1 2 2121 2 1 5 1 7 1 1 and 1, 1, 1 1 2131 1 7 5 24 26 12 6. 7, 5, you have 1 1 1141 7 1 5 Because the value of this determinant is not zero, you can conclude that the three points do not lie on the same line. Moreover, the area of the triangle with vertices square units. at these points is 6 3 1 2 Now try Exercise 31. 333202_0805.qxd 12/5/05 11:05 AM Page 624 624 Chapter 8 Matrices and Determinants The test for collinear points can be adapted to another use. That is, if you are given two points on a rectangular coordinate system, you can find an equation of the line passing through the two points, as follows. y 5 4 (−1, 3) 2 1 (2, 4) −1 1 2 3 4 x FIGURE 8.4 Solution x1, y1 Let for the equation of a line produces 2, 4 x2, y2 and y 4 3 x 2 1 x124 3 1 1 1 0. 1 y13 2 1 1 Two-Point Form of the Equation of a Line An equation of the line passing through the distinct points x2, y2 is given by x x1 x2 1 0. 1 1 y y1 y2 x1, y1 and Example 5 Finding an Equation of a Line Find an equation of the line passing through the two points shown in Figure 8.4. 2, 4 and 1, 3, as 1, 3. Applying the determinant formula To evaluate this determinant, you can expand by cofactors along the first row to obtain the following. 1 1 114 2 1 4 3 0 x11 y13 1110 0 x 3y 10 0 So, an |
equation of the line is x 3y 10 0. Now try Exercise 39. Note that this method of finding the equation of a line works for all lines, including horizontal and vertical lines. For instance, the equation of the vertical line through 2, 2 2, 0 is x 2 2 1 1 and 1 0 y 0 2 4 2x 0 x 2. 333202_0805.qxd 12/5/05 11:05 AM Page 625 Section 8.5 Applications of Matrices and Determinants 625 Cryptography A cryptogram is a message written according to a secret code. (The Greek word kryptos means “hidden.”) Matrix multiplication can be used to encode and decode messages. To begin, you need to assign a number to each letter in the alphabet (with 0 assigned to a blank space), as follows 19 I 10 J 11 K 12 L 13 M 14 N 15 O 16 P 17 Q 18 R 19 S 20 T 21 U 22 V 23 W 24 X 25 Y 26 Z Then the message is converted to numbers and partitioned into uncoded row matrices, each having entries, as demonstrated in Example 6. n Example 6 Forming Uncoded Row Matrices Write the uncoded row matrices of order 1 3 for the message MEET ME MONDAY. Solution Partitioning the message (including blank spaces, but ignoring punctuation) into groups of three produces the following uncoded row matrices. 0 13 5 0 13 15 14 4 1 25 0 5 5 20 13 Note that a blank space is used to fill out the last uncoded row matrix. Now try Exercise 45. To encode a message, use the techniques demonstrated in Section 8.3 to choose an n n A 1 1 1 invertible matrix such as 2 1 1 2 3 4 and multiply the uncoded row matrices by matrices. Here is an example. A (on the right) to obtain coded row Uncoded Matrix Encoding Matrix A Coded Matrix 13 13 26 21 333202_0805.qxd 12/5/05 11:05 AM Page 626 626 Chapter 8 Matrices and Determinants Example 7 Encoding a Message Use the following invertible matrix to encode the message MEET ME MONDAY Solution The coded row matrices are obtained by multiplying each of the uncoded row matrices found in Example 6 by the matrix as follows. A, Uncoded Matrix Encoding Matrix A Coded Matrix 13 20 5 5 0 0 15 14 1 25 |
1 1 1 1 5 1 13 1 13 21 13 26 33 53 12 18 23 42 5 20 24 56 23 77 So, the sequence of coded row matrices is 13 26 21 33 53 12 18 23 42 5 20 56 24 23 77. Finally, removing the matrix notation produces the following cryptogram. 13 26 21 33 53 12 18 23 42 5 20 56 24 23 77 Now try Exercise 47. For those who do not know the encoding matrix decoding the cryptogram found in Example 7 is difficult. But for an authorized receiver who knows the decoding is simple. The receiver just needs to multiply the encoding matrix coded row matrices by (on the right) to retrieve the uncoded row matrices. Here is an example. A1 A, A, 13 26 Coded 211 1 0 10 6 1 A1 13 8 5 1 5 5 Uncoded 333202_0805.qxd 12/5/05 11:05 AM Page 627 s i b r o C © Historical Note During World War II, Navajo soldiers created a code using their native language to send messages between battalions. Native words were assigned to represent characters in the English alphabet, and they created a number of expressions for important military terms, like iron-fish to mean submarine. Without the Navajo Code Talkers, the Second World War might have had a very different outcome. Section 8.5 Applications of Matrices and Determinants 627 Example 8 Decoding a Message Use the inverse of the matrix to decode the cryptogram 13 26 21 33 53 12 18 23 42 5 20 56 24 23 77. A1 by using the techniques demonstrated in Section 8.3. Solution First find is the decoding matrix. Then partition the message into groups of three to form the coded row matrices. Finally, multiply each coded row matrix by (on the right). A1 A1 A1 Decoded Matrix 1 0 1 0 13 26 Coded Matrix Decoding Matrix 21 1 33 53 12 1 18 23 42 1 56 1 77 1 10 6 1 10 6 1 10 6 1 10 6 1 10 6 1 5 20 1 0 1 0 1 0 23 13 20 5 15 1 24 5 0 5 13 0 13 14 4 25 0 So, the message is as follows. 13 5 5 20 0 13 5 0 13 15 14 4 1 25 Now try Exercise 53. W RITING ABOUT MATHEMATICS Cryptography Use your school’s library, the Internet, or some other reference source to research information about another type of cryptography. Write |
a short paragraph describing how mathematics is used to code and decode messages. 333202_0805.qxd 12/5/05 11:05 AM Page 628 628 Chapter 8 Matrices and Determinants 8.5 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The method of using determinants to solve a system of linear equations is called ________ ________. 2. Three points are ________ if the points lie on the same line. 3. The area of a triangle with vertices A x1, y1,, x2, y2 and x3, y3 is given by ________. 4. A message written according to a secret code is called a ________. 5. To encode a message, choose an invertible matrix A (on the right) to obtain ________ row matrices. by A and multiply the ________ row matrices PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–10, use Cramer’s Rule to solve (if possible) the system of equations. 17. y 18. y 4 (0, 4) 6 (1, 6) 1. 3. 5. 7. 9. 2 4 2 4 3x 4y 5x 3y 3x 2y 6x 4y 0.4x 0.8y 1.6 0.2x 0.3y 2.2 4x y z 5 10 1 3 6 11 2x 2y 3z 5x 2y 6z x 2y 3z 2x y z 3x 3y 2z 2. 4. 6. 8. 10. 47 27 17 76 4x 7y x 6y 6x 5y 13x 3y 2.4x 1.3y 4.6x 0.5y 4x 2y 3z 5x 4y z 2x 2y 5z 8x 5y 2z x 2y 2z 3x y z 14.63 11.51 2 16 4 14 10 1 In Exercises 11–14, use a graphing utility and Cramer’s Rule to solve (if possible) the system of equations. 11. 13. 3x 5y 9z 2 5x 9y 17z 4 3x 3y 5z 1 2x y 2z 6 x 2y 3z 0 3x |
2y z 6 12. 14. 7 8 8 2x 2y 2z x 3y 4z x 2y z 2x 3y 5z 4 3x 5y 9z 7 5x 9y 17z 13 In Exercises 15–24, use a determinant and the given vertices of a triangle to find the area of the triangle. 15. y 16. y 5 4 3 2 1 (1, 5) (0, 0) (3, 14, 5) (0, 0) x −1 −2 4 1 (5, −2) −4 −2 2 4 x (−2, −3) (2, −3) (−2, 1) (3, −1) 2 −2 x 4 19. y 20. y (6, 10) 8 4 (−4, −5) −8 x (6, −1) 4 3 2 1 (4, 3) ( 1 20 21. 22. 23. 24. 2, 4, 2, 3, 1, 5 0, 2, 1, 4, 3, 5 3, 5, 2, 6, 3, 5 2, 4, 1, 5, 3, 2 In Exercises 25 and 26, find a value of such that the triangle with the given vertices has an area of 4 square units. y 25. 26. 5, 1, 0, 2, 2, y 4, 2, 3, 5, 1, y In Exercises 27 and 28, find a value of such that the triangle with the given vertices has an area of 6 square units. y 27. 28. x 6 2, 3, 1, 0, 5, 3, 3, y 1, 1, 8, y 333202_0805.qxd 12/5/05 11:05 AM Page 629 Section 8.5 Applications of Matrices and Determinants 629 29. Area of a Region A large region of forest has been infested with gypsy moths. The region is roughly triangular, as shown in the figure. From the northernmost vertex of the region, the distances to the other vertices are 25 miles south and 10 miles east (for vertex ), and 20 miles south and 28 miles east (for vertex ). Use a graphing utility to approximate the number of square miles in this region. C B A W N S E A 20 25 B 10 28 In Exercises 39– 44, use a determinant |
to find an equation of the line passing through the points. 39. 41. 43. 0, 0, 5, 3 4, 3, 2, 1 1 2, 1 2, 3, 5 40. 42. 44. 0, 0, 2, 2 10, 7, 2, 7 2 3, 4, 6, 12 In Exercises 45 and 46, find the uncoded 1 3 row matrices for the message. Then encode the message using the encoding matrix. Message 45. TROUBLE IN RIVER CITY Encoding Matrix In Exercises 47–50, write a cryptogram for the message using the matrix A. C 46. PLEASE SEND MONEY 30. Area of a Region You own a triangular tract of land, as shown in the figure. To estimate the number of square feet in the tract, you start at one vertex, walk 65 feet east and 50 feet north to the second vertex, and then walk 85 feet west and 30 feet north to the third vertex. Use a graphing utility to determine how many square feet there are in the tract of land]. 47. CALL AT NOON 48. ICEBERG DEAD AHEAD 49. HAPPY BIRTHDAY 50. OPERATION OVERLOAD 85 30 W N S E 65 50 52. In Exercises 51–54, use A1 to decode the cryptogram. 51. A 1 3 2 5 11 21 64 112 25 50 29 53 23 46 40 75 55 92 A 5 7 136 58 178 73 90 36 115 49 199 82 120 51 242 101 173 72 70 28 95 38 115 47 2 3 31. In Exercises 31–36, use a determinant to determine whether the points are collinear. 3, 1, 0, 3, 12, 5 2, 1, 4, 4, 6, 3 0, 2, 1, 2.4, 1, 1.6 3, 5, 6, 1, 10, 2 0, 1, 4, 2, 2, 5 2 2, 3, 3, 3.5, 1, 2 34. 36. 32. 35. 33. 2 In Exercises 37 and 38, find collinear. y such that the points are 37. 2, 5, 4, y, 5, 2 38. 6, 2, 5, y, 3, 5 1 0 2 0 1 3 19 9 53. A 1 1 6 9 9 41 1 64 A 3 54. 0 4 38 21 31 4 2 |
5 2 1 3 19 28 9 19 80 25 5 4 112 140 83 19 25 13 72 118 71 20 21 38 35 23 36 42 48 32 76 61 95 333202_0805.qxd 12/5/05 11:05 AM Page 630 630 Chapter 8 Matrices and Determinants In Exercises 55 and 56, decode the cryptogram by using the inverse of the matrix A. Synthesis ] 55. 20 17 12 15 62 143 181 9 59 24 29 65 144 172 56. 13 56 104 1 25 65 61 112 106 17 73 131 11 57. The following cryptogram was encoded with a 2 2 matrix. 13 10 13 15 6 20 40 8 21 1 The last word of the message is _RON. What is the message? 5 10 5 25 5 19 1 16 18 18 Model It 58. Data Analysis: Supreme Court The table shows the numbers of U.S. Supreme Court cases waiting to be tried for the years 2000 through 2002. (Source: Office of the Clerk, Supreme Court of the United States) y Year Number of cases, y 2000 2001 2002 8965 9176 9406 (a) Use the technique demonstrated in Exercises 67–70 in Section 7.3 to create a system of linear equations t 0 for the data. Let corresponding to 2000. represent the year, with t (b) Use Cramer’s Rule to solve the system from part (a) and find the least squares regression parabola y at2 bt c. (c) Use a graphing utility to graph the parabola from part (b). (d) Use the graph from part (c) to estimate when the number of U.S. Supreme Court cases waiting to be tried will reach 10,000. True or False? statement is true or false. Justify your answer. In Exercises 59– 61, determine whether the 59. In Cramer’s Rule, the numerator is the determinant of the coefficient matrix. 60. You cannot use Cramer’s Rule when solving a system of linear equations if the determinant of the coefficient matrix is zero. 61. In a system of linear equations, if the determinant of the coefficient matrix is zero, the system has no solution. 62. Writing At this point in the text, you have learned several methods for solving systems of linear equations. Briefly describe which method(s) you find easiest to use and which method(s) you find most difficult to use. |
Skills Review In Exercises 63–66, use any method to solve the system of equations. 63. 64. 65. 66. 11 16 x 7y 22 5x y 26 3x 8y 2x 12y x 3y 5z 5x y z 4x 2y z 5x 3y 2z 2x 3y z 4x 10y 5z 14 1 11 7 5 37 In Exercises 67 and 68, sketch the region determined by the constraints. Then find the minimum and maximum values of the objective function and where they occur, subject to the constraints. 67. Objective function: z 6x 4y Constraints: 68. Objective function: z 6x 7y Constraints: x ≥ 0 y ≥ 0 x 6y ≤ 30 6x y ≤ 40 x ≥ 0 y ≥ 0 4x 3y ≥ 24 x 3y ≥ 15 333202_080R.qxd 12/5/05 11:08 AM Page 631 8 Chapter Summary What did you learn? Section 8.1 Write matrices and identify their orders (p. 572). Perform elementary row operations on matrices (p. 574). Use matrices and Gaussian elimination to solve systems of linear equations (p. 577). Use matrices and Gauss-Jordan elimination to solve systems of linear equations (p. 579). Section 8.2 Decide whether two matrices are equal (p. 587). Add and subtract matrices and multiply matrices by scalars (p. 588). Multiply two matrices (p. 592). Use matrix operations to model and solve real-life problems (p. 595). Section 8.3 Verify that two matrices are inverses of each other (p. 602). Use Gauss-Jordan elimination to find the inverses of matrices (p. 603). Use a formula to find the inverses of Use inverse matrices to solve systems of linear equations (p. 607. matrices (p. 606). 2 2 Section 8.4 Find the determinants of Find minors and cofactors of square matrices (p. 613). Find the determinants of square matrices (p. 614). matrices (p. 611). 2 2 Section 8.5 Use Cramer’s Rule to solve systems of linear equations (p. 619). Use determinants to find the areas of triangles (p. 622). Use |
a determinant to test for collinear points and to find an equation of a line passing through two points (p. 623). Use matrices to encode and decode messages (p. 625). Chapter Summary 631 Review Exercises 1–8 9, 10 11–24 25–30 31–34 35–48 49–62 63–66 67–70 71–78 79–82 83–94 95–98 99–102 103–106 107–110 111–114 115–120 121–124 333202_080R.qxd 12/8/05 10:40 AM Page 632 632 Chapter 8 Matrices and Determinants 8 Review Exercises 8.1 In Exercises 1–4, determine the order of the matrix. 1. 4 0 5 3. 3 2. 6 2 5 8 0 In Exercises 5 and 6, write the augmented matrix for the system of linear equations. 5. 3x 10y 15 5x 4y 22 6. 8x 7y 4z 12 3x 5y 2z 20 5x 3y 3z 26 In Exercises 7 and 8, write the system of linear equations y, represented by the augmented matrix. (Use variables z, and w, x, if applicable.) 7 0 2 1 2 4 16 21 10 7 8 4 9 10 3 3 5 3 2 12 1 7. 8. 4 9 5 13 1 4 In Exercises 9 and 10, write the matrix in row-echelon form. Remember that the row-echelon form of a matrix is not unique. 9 10. 4 3 2 8 1 10 16 2 12 x, y, 0 0 11. 2 1 0 9 2 0 and 3 2 1 9 1 1 In Exercises 11–14, write the system of linear equations represented by the augmented matrix. Then use backz. substitution to solve the system. (Use variables ) 1 1 1 1 4 10 12. 14. 13. 0 0 0 0 0 0 In Exercises 15–24, use matrices and Gaussian elimination with back-substitution to solve the system of equations (if possible). 16. 2x 5y 2 3x 7y 1 6 9 11 14 15. 17. 18. 19. 20. 21. 22. 23. 24. 2 22 2x 3y 3z 4x 2y 3z 6x 6y 12z 13 12x 9y z 2 5x 4 |
y x y 0.3x 0.1y 0.13 0.2x 0.3y 0.25 0.2x 0.1y 0.07 0.4x 0.5y 0.01 2x 3y z 10 22 2 2x 3y 3z 3 2x y x 2y 6z 2x x 2x 5y 15z 3x y 3z z y 2y 3z 2z 3y z 2z 4 5 6z 2 2x 2y 2x y 2y 3y 4y 1 4 6 3x x 3z z z 4x 2x w 2w 3w w 3 0 2w 0 3 In Exercises 25–28, use matrices and Gauss-Jordan elimination to solve the system of equations. 1 2 4 26. 25. 2x 3y z 5x 4y 2z 4x 2y 8z 1 5x 3y 8z 6 x y 2z 4x 4y 4z 5 2x y 9z 28. 3x y 7z x 3y 4z 5x 2y z 5x 2y z x y 4z 27. 8 15 17 20 34 8 333202_080R.qxd 12/8/05 10:41 AM Page 633 In Exercises 29 and 30, use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system. 29. 30. 44 1 15 58 x 6y 4z w 5x y z 3w 4y z 8w 3x y 5z 2w 4x 12y 2z x 6y 4z x 6y z 2x 10y 2z 20 12 8 10 Review Exercises 633 42. 8 2 0 1 4 6 52 3 6 8 12 0 0 1 12 4 1 8 43. In Exercises 43 and 44, use the matrix capabilities of a graphing utility to evaluate the expression 44. 2 11 3 6 1 0 2 2 7 8 8.2 In Exercises 31–34, find and x y. In Exercises 45–48, solve for X in the equation given 31. 32. 33. 34 12 9 5x 1 9 0 2 0 1 2x 8 4 4y 2 6x 5 4 0 4 3 1 4 44 3 2 6 16 x 10 7 1 5 2y 0 In Exercises 35–38, if possible, find (a) |
(c) and (d) A B, (b) A B, 4A, A 2 3 A 5 A 5 7 11 7 11 35. 36. 37. 38. A 6 5 A 3B 10 8 12 40 30 3 12 40 20 15 B 3 12 B 4 B 0 B 1 4 20 7, 4 8 In Exercises 39–42, perform the matrix operations. If it is not possible, explain why. 39. 7 1 40. 41. 11 7 21 5 6 10 3 14 5 19 1 87 16 2 2 4 0 20 10 A [4 1 3 0 5 2] 45. 47. X 3A 2B 3X 2A B and B [ 1 2 4 2 1 4]. 46. 48. 6X 4A 3B 2A 5B 3X In Exercises 49–52, find 49. 50. 51. A 2 3 A 5 A 5 7 11 7 11, 52. A 6 5 AB, B 3 12 B 4 if possible. 10 8 12 40 30 20 15 12 40 B 4 20 B 1 4 8 7, In Exercises 53–60, perform the matrix operations. If it is not possible, explain why. 53. 54 57. 4 6 56. 55 333202_080R.qxd 12/5/05 11:08 AM Page 634 634 Chapter 8 Matrices and Determinants 31 4 58. 59. 60. In Exercises 61 and 62, use the matrix capabilities of a graphing utility to find the product. 4 11 12 2 4 61. 62 10 2 1 5 3 1 2 2 63. Manufacturing A tire corporation has three factories, each of which manufactures two products. The number of units of product in one day is aij represented by A 80 40 i in the matrix produced at factory 140 80 120 100. j Find the production levels if production is decreased by 5%. 64. Manufacturing A corporation has four factories, each of which manufactures three types of cordless power tools. The number of units of cordless power tools produced at factory in one day is represented by in the matrix j aij A 80 50 90 70 30 60 90 80 100. 40 20 50 Find the production levels if production is increased by 20%. 65. Manufacturing A manufacturing company produces three kinds of computer games that are shipped to two that are warehouses. The number of units of game j in the matrix shipped to warehouse is represented by i aij A 8200 6500 5400 |
. 7400 9800 4800 The price per unit is represented by the matrix B $10.25 $14.50 $17.75. Compute BA and interpret the result. 66. Long-Distance Plans The charges (in dollars per minute) of two long-distance telephone companies for in-state, stateto-state, and international calls are represented by C. Company A C 0.07 0.10 0.28 B 0.095 0.08 0.25 In-state State-to-state International Type of call You plan to use 120 minutes on in-state calls, 80 minutes on state-to-state calls, and 20 minutes on international calls each month. (a) Write a matrix T that represents the times spent on the phone for each type of call. (b) Compute TC and interpret the result. A. B is the inverse of 1 4 8.3 67. 68. 69. 70. 1 2 1, 2 1 0 2 In Exercises 67–70, show that A 4, 7 A 5 11 A 1 A 1 B 2 7 B 2 11, 71. In Exercises 71–74, find the inverse of the matrix (if it exists). 6 5 1 3 2 0 73. 74. 72 In Exercises 75–78, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 75. 77 76. 2 1 1 78. 8 4 1 1 4 3 18 0 2 2 4 6 1 16 8 2 4 1 2 0 1 1 333202_080R.qxd 12/5/05 11:08 AM Page 635 In Exercises 79–82, use the formula below to find the inverse of the matrix, it it exists. In Exercises 83–90, use an inverse matrix to solve (if possible) the system of linear equations. b a] A1 1 c ad bc[ d 2 2 20 6 5 2 8 3 4 3 7 8 10 7 1 3 2 3 10 4 4 5 8 13 10 47 8 5 13 24 x 4y 2x 7y 5x y 9x 2y 3x 10y 5x 17y 4x 2y 19x 9y 3x 2y z x 4y 2z 2x y 2z 3x y 5z 2x 9y 5z x 5y 4z x 4y z y z x y 2z 5x y z x y 6 |
z 8x 4y z 6 1 7 12 25 10 13 11 0 14 8 44 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. In Exercises 91–94, use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. 91. 92. 93. x 2y 1 3x 4y 5 x 3y 6x 2y 3x 3y 4z y z 4x 3y 4z 23 18 2 1 1 Review Exercises 635 94. x 3y 2z 2x 7y 3z x y 3z 8 19 3 In Exercises 95–98, find the determinant of the 95. 8.4 matrix. 8 2 9 7 50 10 14 12 98. 96. 97. 5 4 11 4 30 5 24 15 99. 100. 6 4 1 4 In Exercises 99–102, find all (a) minors and (b) cofactors of the matrix 101. 102. 3 5 1 4 9 2 6 4 In Exercises 103–106, find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. 103. 104. 2 5 6 5 2 4 3 106. 5 0 6 0 105 333202_080R.qxd 12/5/05 11:08 AM Page 636 636 Chapter 8 Matrices and Determinants In Exercises 107–110, use Cramer’s Rule to solve (if 8.5 possible) the system of equations. In Exercises 123 and 124, decode the cryptogram by using the inverse of the matrix 107. 109. 6 23 5x 2y 11x 3y 2x 3y 5z 4x y z x 4y 6z 11 3 15 108. 110. 7 37 3x 8y 9x 5y 5x 2y z 3x 3y z 2x y 7z 15 7 3 In Exercises 111–114, use a determinant and the given vertices of a triangle to find the area of the triangle. A [5 10 8 4 7 6 3 6 5]. 123. 124. 145 105 92 264 188 160 23 16 15 5 11 2 370 265 225 57 48 33 32 15 20 245 171 147 129 84 78 9 8 5 159 118 100 219 152 133 370 265 225 105 84 63 111. y 112. |
y Synthesis 8 6 4 2 −2 (5, 8) (5, 0) (1, 0) 4 6 8 113. (−2, 3) y 6 2 (0, 5) −4 −2 −2 −4 2 4 (1, −4) x x (0, 6) (4, 0) 6 2 −4 −2 (−4, 0) 2 4 114. y 3 2 1 3 2( (, 1 (4, 2) 1 2 3 1 4, − 2 ( ( x x In Exercises 115 and 116, use a determinant to determine whether the points are collinear. 1, 7, 3, 9, 3, 15 0, 5, 2, 6, 8, 1 116. 115. In Exercises 117–120, use a determinant to find an equation of the line passing through the points. 117. 119. 4, 0, 4, 4 5 2, 1 2, 3, 7 118. 120. 2, 5, 6, 1 0.8, 0.2, 0.7, 3.2 In Exercises 121 and 122, find the uncoded row matrices for the message. Then encode the message using the encoding matrix. 1 3 Message 121. LOOK OUT BELOW 122. RETURN TO BASE Encoding Matrix True or False? In Exercises 125 and 126, determine whether the statement is true or false. Justify your answer. 125. It is possible to find the determinant of a a12 a22 c2 a13 a23 126. a33 a31 4 5 matrix. c3 a13 a23 c3 a12 a22 c2 a32 a13 a23 a33 a11 a21 c1 a11 a21 a31 a11 a21 c1 a12 a22 a32 127. Under what conditions does a matrix have an inverse? 128. Writing What is meant by the cofactor of an entry of a matrix? How are cofactors used to find the determinant of the matrix? 129. Three people were asked to solve a system of equations using an augmented matrix. Each person reduced the matrix to row-echelon form. The reduced matrices were 1 0, 3 1 2 1 1 0 and Can all three be right? Explain. 130. Think About It Describe the row-echelon form of an augmented matrix that corresponds to a system of linear equations that has a unique solution. 131. Sol |
ve the equation for. 2 3 8 0 5 333202_080R.qxd 12/5/05 11:08 AM Page 637 8 Chapter Test Chapter Test 637 Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. In Exercises 1 and 2, write the matrix in reduced row-echelon form. 1. Write the augmented matrix corresponding to the system of equations and solve the system. 4x 3y 2z x y 2z 3x y 4z 14 5 8 4. Find (a) A 5 4 A B, (b) 3, 4 4 A, (c) B 4 4 3A 2B, 1 0 and (d) AB (if possible). In Exercises 5 and 6, find the inverse of the matrix (if it exists). 5. 6 10 4 5 6. Use the result of Exercise 5 to solve the system. 6x 4y 10 10x 5y 20 y 6 4 (−5, 0) −4 −2 −2 (4, 4) (3, 2) 2 4 x In Exercises 8–10, evaluate the determinant of the matrix. 8. 9 13 4 16 9. 5 2 8 13 4 6 5 10 In Exercises 11 and 12, use Cramer’s Rule to solve (if possible) the system of equations. 11. 7x 6y 2x 11y 9 49 12. 6x y 2z 2x 3y z 4x 4y z 4 10 18 FIGURE FOR 13 13. Use a determinant to find the area of the triangle in the figure. 14. Find the uncoded 1 3 row matrices for the message KNOCK ON WOOD. Then encode the message using the matrix below 15. One hundred liters of a 50% solution is obtained by mixing a 60% solution with a 20% solution. How many liters of each solution must be used to obtain the desired mixture? 333202_080R.qxd 12/5/05 11:08 AM Page 638 Proofs in Mathematics Proofs without words are pictures or diagrams that give a visual understanding of why a theorem or statement is true. They can also provide a starting point for determinant is the writing a formal proof. The following proof shows that a area of a parallelogram. 2 2 (0, d) (a, b + d) |
(a + c, b + d) (a, d) (a + c, d) (a, b) (0, 0) (a, 0) a c b d ad bc The following is a color-coded version of the proof along with a brief expla- nation of why this proof works. (0, d) (a, b + d) (a + c, b + d) (a, d) (a + c, d) (a, b) (0, 0) (a, 0) a c b d ad bc Area of Area of yellow Area of blue Area of orange Area of pink Area of white quadrilateral Area of Area of orange quadrilateral Area of pink Area of green Area of Area of white quadrilateral Area of blue Area of Area of green quadrilateral Area of Area of yellow From “Proof Without Words” by Solomon W. Golomb, Mathematics Magazine, March 1985. Vol. 58, No. 2, pg. 107. Reprinted with permission. 638 333202_080R.qxd 12/5/05 11:08 AM Page 639 P.S. Problem Solving This collection of thought-provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. The columns of matrix T vertices of a triangle. Matrix A 0 T 1 1 1 1 0 show the coordinates of the A is a transformation matrix. 2 4 3 2 (a) Find AT and AAT. Then sketch the original triangle and the two transformed triangles. What transformation does A represent? 4. Let A 1 2 (a) Show that. 2 1 A2 2A 5I O, where I is the identity matrix of order 2. A1 1 5 (b) Show that 2I A. (c) Show in general that for any square matrix satisfying A2 2A 5I O (b) Given the triangle determined by describe the transformation process that produces the triangle determined by and then the triangle determined by AAT, AT T. 2. The matrices show the number of people (in thousands) who lived in each region of the United States in 2000 and the number of people (in thousands) projected to live in each region in 2015. The regional populations are separated into three age categories. (Source: U.S. Census Bureau) 0–17 13,049 16,646 25,569 4,935 12,098 0 |
–17 12,589 15,886 25,916 5,226 14,906 Northeast Midwest South Mountain Pacific Northeast Midwest South Mountain Pacific 2000 18–64 33,175 39,486 62,235 11,210 28,036 2015 18–64 34,081 41,038 68,998 12,626 33,296 65 + 7,372 8,263 12,437 2,031 4,893 65 + 8,165 10,101 17,470 3,270 6,565 (a) The total population in 2000 was 281,435,000 and the projected total population in 2015 is 310,133,000. Rewrite the matrices to give the information as percents of the total population. (b) Write a matrix that gives the projected change in the percent of the population in each region and age group from 2000 to 2015. (c) Based on the result of part (b), which region(s) and age group(s) are projected to show relative growth from 2000 to 2015? 3. Determine whether the matrix is idempotent. A square A2 A. matrix is idempotent if (ac) (b) (d is given by the inverse of A1 1 5 2I A. 5. Two competing companies offer cable television to a city with 100,000 households. Gold Cable Company has 25,000 subscribers and Galaxy Cable Company has 30,000 subscribers. (The other 45,000 households do not subscribe.) The percent changes in cable subscriptions each year are shown in the matrix below. Percent Changes From Gold 0.70 0.20 0.10 From From NonGalaxy subscriber 0.15 0.80 0.05 0.15 0.15 0.70 Percent Changes To Gold To Galaxy To Nonsubscriber (a) Find the number of subscribers each company will have in 1 year using matrix multiplication. Explain how you obtained your answer. (b) Find the number of subscribers each company will have in 2 years using matrix multiplication. Explain how you obtained your answer. (c) Find the number of subscribers each company will have in 3 years using matrix multiplication. Explain how you obtained your answer. (d) What is happening to the number of subscribers to each company? What is happening to the number of nonsubscribers? 6. Find such that the matrix is equal to its own inverse. x A 3 2 x 3 7. Find such that the matrix is singular. x 3 x A 4 2 8 |
. Find an example of a singular 2 2 matrix satisfying A2 A. 639 333202_080R.qxd 12/5/05 11:08 AM Page 640 10. Verify the following equation. 9. Verify the following equation. a a2 a a3 1 1 1 b b2 1 b b3 1 c c2 a bb cc a c3 a bb cc aa b c a ax2 bx 11. Verify the following equation. 12. Use the equation given in Exercise 11 as a model to find a determinant that is equal to ax3 bx2 cx d. 13. The atomic masses of three compounds are shown in the table. Use a linear system and Cramer’s Rule to find the atomic masses of sulfur (S), nitrogen (N), and fluorine (F). Compound Formula Atomic mass Tetrasulphur tetranitride Sulfur hexafluoride Dinitrogen tetrafluoride S4N4 SF6 N2F4 184 146 104 14. A walkway lighting package includes a transformer, a certain length of wire, and a certain number of lights on the wire. The price of each lighting package depends on the length of wire and the number of lights on the wire. Use the following information to find the cost of a transformer, the cost per foot of wire, and the cost of a light. Assume that the cost of each item is the same in each lighting package. • A package that contains a transformer, 25 feet of wire, and 5 lights costs $20. • A package that contains a transformer, 50 feet of wire, and 15 lights costs $35. • A package that contains a transformer, 100 feet of wire, and 20 lights costs $50. 15. The transpose of a matrix, denoted is formed by writing its columns as rows. Find the transpose of each ABT BTAT. matrix and verify that AT 640 16. Use the inverse of matrix A to decode the cryptogram 23 13 24 14 56 38 41 53 34 37 31 41 34 17 116 13 85 28 32 11 16 63 25 8 20 1 22 17 29 3 61 40 6 17. A code breaker intercepted the encoded message below. 35 28 45 42 10 Let A1 w y 38 30 18 75 55 18 2 35 2 30 22 81 21 60 15. x z (a) You know that 45 35 A1 10 15 and that 38 30 A1 8 14, is the inverse of |
the encoding matrix Write and solve two systems y, of equations to find A. x,w, where A1 and z. (b) Decode the message. 18. Let. Use a graphing utility to find Make a conjecture about the determinant of the inverse of a matrix. Compare A1 A1. with n n 19. Let A zero. Find be an A. 20. Consider matrices of the form matrix each of whose rows adds up to A 0 0 0 0 0 a12 0 0 0 0 a13 a23 0 0 0 a14 a24 a34 0 0.................. a1n a2n a3n an1n 0 (a) Write a of A. 2 2 matrix and a 3 3 matrix in the form (b) Use a graphing utility to raise each of the matrices to higher powers. Describe the result. (c) Use the result of part (b) to make a conjecture about matrix. Use a graphing is a A A 4 4 if powers of utility to test your conjecture. (d) Use the results of parts (b) and (c) to make a conjecture is an about powers of matrix. AA if n n 99 333202_0900.qxd 12/5/05 11:26 AM Page 641 Sequences, Series, and Probability 9.1 9.2 9.3 Sequences and Series Arithmetic Sequences and Partial Sums Geometric Sequences and Series 9.4 Mathematical Induction 9.5 9.6 9.7 The Binomial Theorem Counting Principles Probability Poker has become a popular card game in recent years.You can use the probability theory developed in this chapter to calculate the likelihood of getting different poker hands AT I O N S Sequences, series, and probability have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. • Federal Debt, Exercise 111, page 651 • Data Analysis: Tax Returns, • Lottery, Exercise 61, page 682 Exercise 65, page 700 • Falling Object, • Child Support, Exercises 87 and 88, page 661 Exercise 80, page 690 • Multiplier Effect, • Poker Hand Exercises 113–116, page 671 Exercise 57, page 699 • Defective Units, Exercise 47, page 711 • Population Growth, Exercise 139, page 718 641 333202_0901.qxd 12/ |
5/05 11:28 AM Page 642 642 Chapter 9 Sequences, Series, and Probability 9.1 Sequences and Series What you should learn • Use sequence notation to write the terms of sequences. • Use factorial notation. • Use summation notation to write sums. • Find the sums of infinite series. • Use sequences and series to model and solve real-life problems. Why you should learn it Sequences and series can be used to model real-life problems. For instance, in Exercise 109 on page 651, sequences are used to model the number of Best Buy stores from 1998 through 2003. Scott Olson /Getty Images The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter. Sequences In mathematics, the word sequence is used in much the same way as in ordinary English. Saying that a collection is listed in sequence means that it is ordered so that it has a first member, a second member, a third member, and so on. Mathematically, you can think of a sequence as a function whose domain is the set of positive integers. f 2 a2, f1 a1, f 3 a3, f 4 a4,..., f n an,... Rather than using function notation, however, sequences are usually written using subscript notation, as indicated in the following definition. Definition of Sequence An infinite sequence is a function whose domain is the set of positive integers. The function values a1, a2, a3, a4,..., an,... are the terms of the sequence. If the domain of the function consists of the first positive integers only, the sequence is a finite sequence. n On occasion it is convenient to begin subscripting a sequence with 0 instead of 1 so that the terms of the sequence become a0, a1, a2, a3,.... Example 1 Writing the Terms of a Sequence Write the first four terms of the sequences given by 3n 2 a. an Solution b. an 3 1n. a. The first four terms of the sequence given by an 3n 2 are 31 2 1 32 2 4 33 2 7 34 2 10. a1 a2 a3 a4 1st term 2nd term 3rd term 4th term 3 1n are an a1 b. The first four terms of the sequence given by 3 11 3 1 2 3 12 3 1 4 3 13 3 |
1 2 3 14 3 1 4. a2 a3 a4 1st term 2nd term 3rd term 4th term Now try Exercise 1. 333202_0901.qxd 12/5/05 11:28 AM Page 643 Section 9.1 Sequences and Series 643 Example 2 A Sequence Whose Terms Alternate in Sign Write the first five terms of the sequence given by an 1n 2n 1. Solution The first five terms of the sequence are as follows. 1 2 1 1 a1 1st term 11 21 1 12 22 1 13 23 1 14 24 1 15 25 1 a2 a3 a4 a5 10 1 1 9 2nd term 3rd term 4th term 5th term Now try Exercise 17. Simply listing the first few terms is not sufficient to define a unique th term must be given. To see this, consider the following n sequence—the sequences, both of which have the same first three terms 16, 1 15,..., 1 2n,...,..., 6 n 1n2 n 6,... Example 3 Finding the nth Term of a Sequence Write an expression for the apparent th term of each sequence. an n 2, 5, 10, 17,... a. 1, 3, 5, 7,... b. Solution a. n: 1 2 3 4 Terms: 1 3 5 7 Apparent pattern: Each term is 1 less than twice which implies that... n... an n, an 2n 1. b. n: 1 2 5 3 10 4 17... n... an Terms: 2 Apparent pattern: The terms have alternating signs with those in the even positions being negative. Each term is 1 more than the square of which implies that n, Exploration Write out the first five terms of the sequence whose th term is 1n1 2n 1 an n. Are they the same as the first five terms of the sequence in Example 2? If not, how do they differ? Te c h n o l o g y To graph a sequence using a graphing utility, set the mode to sequence and dot and enter the sequence. The graph of the sequence in Example 3(a) is shown below. You can use the trace feature or value feature to identify the terms. 11 0 0 5 1n1n2 1 an Now try Exercise 37. 333202_0901.qxd 12/5/05 11 |
:28 AM Page 644 644 Chapter 9 Sequences, Series, and Probability Some sequences are defined recursively. To define a sequence recursively, you need to be given one or more of the first few terms. All other terms of the sequence are then defined using previous terms. A well-known example is the Fibonacci sequence shown in Example 4. Example 4 The Fibonacci Sequence: A Recursive Sequence The Fibonacci sequence is defined recursively, as follows. ak2 ak1, Write the first six terms of this sequence. k ≥ 2 where 1, 1, a0 a1 ak a4 The subscripts of a sequence make up the domain of the sequence and they serve to identify the location of a term within the sequence. For examis the fourth term of the ple, an is the nth term sequence, and of the sequence. Any variable can be used as a subscript. The most commonly used variable subscripts in sequence and series notation are and n. k, i, j, Solution a0 1 1 a22 a32 a42 a52 a1 a2 a3 a4 a5 a21 a31 a41 a51 a0 a1 a2 a3 a1 a2 a3 a4 Now try Exercise 51. 0th term is given. 1st term is given. Use recursion formula. Use recursion formula. Use recursion formula. Use recursion formula. Factorial Notation Some very important sequences in mathematics involve terms that are defined with special types of products called factorials. Definition of Factorial If n is a positive integer, n factorial is defined as n. As a special case, zero factorial is defined as 0! 1. that for the first several nonnegative integers. Notice n! Here are some values of is 1 by definition. 0! 0! 1 1! 1 2! 1 2 2 3! 1 2 3 6 4! 1 2 3 4 24 5! 1 2 3 4 5 120 The value of does not have to be very large before the value of extremely large. For instance, 10! 3,628,800. n n! becomes 333202_0901.qxd 12/5/05 11:28 AM Page 645 Section 9.1 Sequences and Series 645 Factorials follow the same conventions for order of operations as do expo- nents. For instance, 2n! 2n! 21 2 3 4.. |
. n whereas 2n! 1 2 3 4... 2n. Example 5 Writing the Terms of a Sequence Involving Factorials Write the first five terms of the sequence given by an. 2n n! n 0. Begin with Then graph the terms on a set of coordinate axes. Solution a0 a1 a 2 a3 a4 20 0! 21 1! 22 2! 23 3! 24 4 16 24 2 3 0th term 1st term 2nd term 3rd term 4th term an 4 3 2 1 1 2 3 4 n FIGURE 9.1 Figure 9.1 shows the first five terms of the sequence. Now try Exercise 59. When working with fractions involving factorials, you will often find that the fractions can be reduced to simplify the computations. Example 6 Evaluating Factorial Expressions Evaluate each factorial expression. a. 8! 2! 6! Solution b. 2! 6! 3! 5! c. n! n 1! a. b. c. 8! 2! 6! 2! 6! 3! 5! n! n 1 28 2 6 3 n Now try Exercise 69. Note in Example 6(a) that you can simplify the computation as follows. 8! 2! 6! 8 7 6! 2! 6! 8 7 2 1 28 333202_0901.qxd 12/5/05 11:28 AM Page 646 646 Chapter 9 Sequences, Series, and Probability Te c h n o l o g y Summation Notation Most graphing utilities are able to sum the first n terms of a sequence. Check your user’s guide for a sum sequence feature or a series feature. Summation notation is an instruction to add the terms of a sequence. From the definition at the right, the upper limit of summation tells you where to end the sum. Summation notation helps you generate the appropriate terms of the sequence prior to finding the actual sum, which may be unclear. There is a convenient notation for the sum of the terms of a finite sequence. It is called summation notation or sigma notation because it involves the use of the uppercase Greek letter sigma, written as. Definition of Summation Notation The sum of the first n terms of a sequence is represented by n i1 ai a1 a2 a3 a4... an i is called the index of summation, where summation, and 1 is the |
lower limit of summation. n is the upper limit of Example 7 Summation Notation for Sums Find each sum. a. 5 i1 3i b. 6 k3 1 k2 c. 8 i0 1 i! Solution 5 a. i1 3i 31 32 33 34 35 31 2 3 4 5 315 45 b. 6 k3 1 k 2 1 32 1 42 1 52 1 62 10 17 26 37 90 c. 8 i0 1 i! 1 0! 1 1! 1 2! 1 6 1 3! 1 24 1 4! 1 120 1 1 6! 5! 1 720 1 7! 1 1 8! 1 5040 40,320 1 1 1 2 2.71828 For this summation, note that the sum is very close to the irrational number e 2.718281828. It can be shown that as more terms of the sequence whose n th term is are added, the sum becomes closer and closer to 1n! e. Now try Exercise 73. In Example 7, note that the lower limit of a summation does not have to be 1. Also note that the index of summation does not have to be the letter For instance, in part (b), the letter is the index of summation. i. k 333202_0901.qxd 12/5/05 11:28 AM Page 647 Section 9.1 Sequences and Series 647 Properties of Sums Variations in the upper and lower limits of summation can produce quite different-looking summation notations for the same sum. For example, the following two sums have the same terms. 3 32i 321 22 23 i1 2 i0 32i1 321 22 23 1. 3. n i1 c cn, c is a constant. 2. n i1 cai cn i1 ai, c is a constant. n i1 ai bi n i1 ai n i1 bi 4. n i1 ai bi n ai i1 n bi i1 For proofs of these properties, see Proofs in Mathematics on page 722. Series Many applications involve the sum of the terms of a finite or infinite sequence. Such a sum is called a series. Definition of Series Consider the infinite sequence a1, a2, a3,..., ai,.... 1. The sum of the first n terms of the sequence is called a finite series or the nth partial sum of the |
sequence and is denoted by a1 a2 a3... an n i1 ai. 2. The sum of all the terms of the infinite sequence is called an infinite series and is denoted by a1 a2 a3... ai... ai. i1 Example 8 Finding the Sum of a Series For the series i1 3 10i, find (a) the third partial sum and (b) the sum. Solution a. The third partial sum is 3 i1 3 10i 3 101 3 102 3 103 0.3 0.03 0.003 0.333. b. The sum of the series is i1 3 10i... 3 105 3 103 3 101 3 102 3 104 0.3 0.03 0.003 0.0003 0.00003.. 0.33333... 1 3.. Now try Exercise 99. 333202_0901.qxd 12/5/05 11:28 AM Page 648 648 Chapter 9 Sequences, Series, and Probability Application Sequences have many applications in business and science. One such application is illustrated in Example 9. Example 9 Population of the United States For the years 1980 to 2003, the resident population of the United States can be approximated by the model an 226.9 2.05n 0.035n2, n 0, 1,..., 23 an is the population (in millions) and n 0 where corresponding to 1980. Find the last five terms of this finite sequence, which represent the U.S. population for the years 1999 to 2003. (Source: U.S. Census Bureau) represents the year, with n Solution The last five terms of this finite sequence are as follows. a19 a20 a21 a22 a23 226.9 2.0519 0.035192 278.5 226.9 2.0520 0.035202 281.9 226.9 2.0521 0.035212 285.4 226.9 2.0522 0.035222 288.9 226.9 2.0523 0.035232 292.6 Now try Exercise 111. 1999 population 2000 population 2001 population 2002 population 2003 population Exploration 3 3 3 A cube is created using 27 unit cubes (a unit cube has a length, width, and height of 1 unit) and only the faces of each cube that are visible are painted blue (see Figure 9.2). Complete the table below to |
determine how cube have 0 blue faces, 1 blue face, 2 blue many unit cubes of the faces, and 3 blue faces. Do the same for a cube, and a pattern do you observe in the table? Write a formula you could use to determine the column values for an cube and add your results to the table below. What type of cube, a cube. Number of blue cube faces 3 3 3 0 1 2 3 FIGURE 9.2 333202_0901.qxd 12/5/05 11:28 AM Page 649 9.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. Section 9.1 Sequences and Series 649 VOCABULARY CHECK: Fill in the blanks. 1. An ________ ________ is a function whose domain is the set of positive integers. 2. The function values 3. A sequence is a ________ sequence if the domain of the function consists of the first positive integers. 4. If you are given one or more of the first few terms of a sequence, and all other terms of the sequence are are called the ________ of a sequence. a1, a2, a3, a4,... n defined using previous terms, then the sequence is said to be defined ________. 5. If n is a positive integer, n ________ is defined as n. 6. The notation used to represent the sum of the terms of a finite sequence is ________ ________ or sigma notation. 7. For the sum in ai, the ________ limit of summation. i1 is called the ________ of summation, n is the ________ limit of summation, and 1 is 8. The sum of the terms of a finite or infinite sequence is called a ________. 9. The ________ ________ ________ of a sequence is the sum of the first n terms of the sequence. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–22, write the first five terms of the sequence. n (Assume that begins with 1.) In Exercises 27–32, use a graphing utility to graph the first 10 terms of the sequence. (Assume that begins with 1.) n 1. |
3. 5. an an an 7. an 9. an 11. an 13. an 15. an 17. an 19. 21. an an 3n 1 2n 2n n 2 n 6n 3n 2 1 1 1n n 2 1 3n 1 n32 1n n2 2 3 nn 1n 2 2. 4. 6. an an an 8. an 10. an 5n 3n2 n 4 2n2 1 12. an 1 1n 14. an 16. an 18. an 20. 22. an an 2n 3n 10 n23 1n n n 1 0.3 nn2 6 23. an a25 In Exercises 23–26, find the indicated term of the sequence. 1n1nn 1 4n2 n 3 nn 1n 2 1n3n 2 4n 2n2 3 an a16 26. 25. 24. an an a11 a13 27. an 29. an 31. an 3 4 n 160.5n1 2n n 1 28. an 30. an 32. an 2 4 n 80.75n1 n2 n2 2 In Exercises 33–36, match the sequence with the graph of its first 10 terms. [The graphs are labeled (a), (b), (c), and (d).] (a) (b) an an 2 4 6 8 10 10 8 6 4 2 (c) an 10 8 6 4 2 10 8 6 4 2 (d) an 10 10 n n 2 4 6 8 10 2 4 6 8 10 33. an 35. an 8 n 1 40.5n1 34. an 36. an 8n n 1 4n n! 333202_0901.qxd 12/5/05 11:28 AM Page 650 650 Chapter 9 Sequences, Series, and Probability In Exercises 37–50, write an expression for the apparent th term of the sequence. (Assume that begins with 1.) n n 37. 1, 4, 7, 10, 13,... 39. 41. 43. 45. 0, 3, 8, 15, 24, 4, 7, 6 3, 4 1, 3 5, 5 9,... 9, 1 4, 1 1, 1 1 16, 25,... 2 47. 1, 1, 1, 1, 1,... 38. 40. 42. 44. 46 |
. 48. 49. 50. 1 1 1 1 1, 1 1 2, 1 3 2, 1 1 4, 1 7 3, 1 1 8, 1 15 4, 1 1 16, 1 31 3, 7, 11, 15, 19,... 2, 4, 6, 8, 10,... 1 1 16,... 2, 1 3, 2 81,... 24, 1 1, 1 120,... 23 24 25 6 120 24 1 4, 1 8, 27, 8 9, 4 6, 1 2, 1 22 2 5,... 32,... 1, 2,,,,,... In Exercises 51–54, write the first five terms of the sequence defined recursively. 51. 52. 53. 54. 28, 15, 3, 32, a1 a1 a1 a1 4 3 1 ak1 ak1 ak1 ak1 ak ak 2ak 1 2ak In Exercises 55–58, write the first five terms of the sequence defined recursively. Use the pattern to write the th term of the sequence as a function of. (Assume that begins with 1.) n n n 55. 56. 57. 58. 6, 25, 81, 14, a1 a1 a1 a1 ak1 ak1 ak1 ak1 2 5 ak ak 1 3ak 2ak In Exercises 59–64, write the first five terms of the sequence. (Assume that begins with 0.) n 59. an 3n n! 61. an 63. an 1 n 1! 12n 2n! n! n n2 60. an 62. an n 1! 12n1 2n 1! 64. an In Exercises 65–72, simplify the factorial expression. 65. 67. 69. 71. 4! 6! 10! 8! n 1! n! 2n 1! 2n 1! 66. 68. 70. 72. 5! 8! 25! 23! n 2! n! 3n 1! 3n! In Exercises 73–84, find the sum. 73. 75. 77. 79. 81. 83. 5 i1 2i 1 4 k1 10 4 i0 i 2 3 k0 1 k2 1 5 k2 k 12k 3 4 i1 2i 74. 76. 78. 80. 82. 84. 6 |
i1 3i 1 5 5 k1 5 i0 2i 2 5 j3 1 j 2 3 4 i1 i 12 i 13 4 j0 2 j In Exercises 85–88, use a calculator to find the sum. 86. 10 j1 3 j 1 85. 87. 88. 6 j1 24 3j 4 k0 4 k0 1k k 1 1k k! In Exercises 89–98, use sigma notation to write the sum. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98.... 1 39... 22 1 32 5 1 33 5 1 31 27 81 243 729 1 1 2 1 1 1 12 32 22 1 2 4... 1 128 1 8 1 4 6... 1 202... 1 2... 1 6 10 12 1 15 3 2 8 6 1 42 1 3 5 31 64 120 32 7 16 6 8 15 32 24 16 720 64 In Exercises 99–102, find the indicated partial sum of the series. 99. 101. 51 2 i i1 Fourth partial sum n 41 2 n1 Third partial sum 100. 102. 21 3 i i1 Fifth partial sum n 81 4 n1 Fourth partial sum 333202_0901.qxd 12/5/05 11:28 AM Page 651 In Exercises 103–106, find the sum of the infinite series. Model It (co n t i n u e d ) Section 9.1 Sequences and Series 651 103. 104. 105. 106. k1 i1 i1 k1 6 1 10 i k 1 10 7 1 10 k 2 1 10 i 107. Compound Interest A deposit of $5000 is made in an account that earns 8% interest compounded quarterly. The balance in the account after quarters is given by n An 50001 0.08 4 n, n 1, 2, 3,.... (a) Write the first eight terms of this sequence. (b) Find the balance in this account after 10 years by finding the 40th term of the sequence. 108. Compound Interest A deposit of $100 is made each month in an account that earns 12% interest compounded monthly. The balance in the account after months is given by n 1001011.01n 1, n 1, 2, 3,.... An (a) Write the first six terms of this sequence. ( |
b) Find the balance in this account after 5 years by finding the 60th term of the sequence. (c) Find the balance in this account after 20 years by finding the 240th term of the sequence. Model It 109. Data Analysis: Number of Stores The table shows of Best Buy stores for the years 1998 the numbers to 2003. an (Source: Best Buy Company, Inc.) Year 1998 1999 2000 2001 2002 2003 Number of an stores, 311 357 419 481 548 608 (a) Use the regression feature of a graphing utility to n find a linear sequence that models the data. Let corresponding to represent the year, with 1998. n 8 (b) Use the regression feature of a graphing utility to find a quadratic sequence that models the data. (c) Evaluate the sequences from parts (a) and (b) for n 8, 9,..., 13. Compare these values with those shown in the table. Which model is a better fit for the data? Explain. (d) Which model do you think would better predict the number of Best Buy stores in the future? Use the model you chose to predict the number of Best Buy stores in 2008. 110. Medicine The numbers (in thousands) of AIDS cases reported from 1995 to 2003 can be approximated by the model an 0.0457n3 0.352n2 9.05n 121.4, an n 5, 6,..., 13 n where is the year, with (Source: U.S. Centers Prevention) n 5 corresponding to 1995. for Disease Control and (a) Find the terms of this finite sequence. Use the statistical plotting feature of a graphing utility to construct a bar graph that represents the sequence. (b) What does the graph in part (a) say about reported cases of AIDS? 111. Federal Debt From 1990 to 2003, the federal debt of the United States rose from just over $3 trillion to almost $7 (in billions of dollars) from trillion. The federal debt 1990 to 2003 is approximated by the model an 2.7698n3 61.372n2 600.00n 3102.9, an n 0, 1,..., 13 n where (Source: U.S. Office of Management and Budget) is the year, with corresponding to 1990. n 0 (a) Find the terms of this finite sequence. Use the statistical plotting feature of a graphing utility to construct a bar graph that represents |
the sequence. (b) What does the pattern in the bar graph in part (a) say about the future of the federal debt? 333202_0901.qxd 12/5/05 11:28 AM Page 652 652 Chapter 9 Sequences, Series, and Probability 112. Revenue The revenues an (in millions of dollars) for Amazon.com for the years 1996 through 2003 are shown in the figure. The revenues can be approximated by the model an 46.609n2 119.84n 1125.8, n 6, 7,..., 13 n is the year, with n 6 where corresponding to 1996. Use this model to approximate the total revenue from 1996 through 2003. Compare this sum with the result of (Source: adding the revenues shown in the figure. Amazon.com) an e u n e v e R 6000 5000 4000 3000 2000 1000 ) ( Synthesis 6 7 9 11 8 Year (6 ↔ 1996) 10 n 12 13 True or False? In Exercises 113 and 114, determine whether the statement is true or false. Justify your answer. 118. Find the arithmetic mean of the following prices per gallon for regular unleaded gasoline at five gasoline stations in a city: $1.899, $1.959, $1.919, $1.939, and $1.999. Use the statistical capabilities of a graphing utility to verify your result. 119. Proof Prove that n i1 xi x 0. 120. Proof Prove that n i1 xi x 2 n i1 x 2 i 1 n n i1 xi2. In Exercises 121–124, find the first five terms of the sequence. 121. an 123. an xn n! 1n x2n 2n! Skills Review 122. an 124. an 1n x2n1 2n 1 1n x2n1 2n 1! In Exercises 125–128, determine whether the function has an inverse function. If it does, find its inverse function. 125. f x 4x 3 127. hx 5x 1 126. 128. gx 3 x f x x 12 In Exercises 129–132, find (a) and (d) A B, (b) 4B 3A, (c) AB, 113. 4 i2 2i 4 i 2 24 i1 i1 i i1 114. 4 2 j 6 2 |
j2 j1 j3 Fibonacci Sequence Fibonacci sequence. (See Example 4.) In Exercises 115 and 116, use the 115. Write the first 12 terms of the Fibonacci sequence the first 10 terms of the sequence given by an and bn an1 an, n ≥ 1. 129. 130. 131. 116. Using the definition for bn in Exercise 115, show that bn 132. can be defined recursively by 4 3 12 11 BA. A 6 3 A 10 bn 1 1 bn1. Arithmetic Mean In Exercises 117–120, use the following definition of the arithmetic mean n measurements x2, n of a set of..., xn. x1, x3, xi x x 1 n i1 117. Find the arithmetic mean of the six checking account balances $327.15, $785.69, $433.04, $265.38, $604.12, and $590.30. Use the statistical capabilities of a graphing utility to verify your result. In Exercises 133–136, find the determinant of the matrix. 8 15 A 2 12 5 7 133. 134. 135. A 3 1 A 3 136. A 16 0 4 9 2 4 4 7 9 11 8 1 6 5 3 1 10 3 12 2 2 7 3 1 333202_0902.qxd 12/5/05 11:29 AM Page 653 Section 9.2 Arithmetic Sequences and Partial Sums 653 9.2 Arithmetic Sequences and Partial Sums What you should learn • Recognize, write, and find the nth terms of arithmetic sequences. • Find nth partial sums of arithmetic sequences. • Use arithmetic sequences to model and solve real-life problems. Arithmetic Sequences A sequence whose consecutive terms have a common difference is called an arithmetic sequence. Definition of Arithmetic Sequence A sequence is arithmetic if the differences between consecutive terms are the same. So, the sequence Why you should learn it a1, a2, a3, a4,..., an,... Arithmetic sequences have practical real-life applications. For instance, in Exercise 83 on page 660, an arithmetic sequence is used to model the seating capacity of an auditorium. is arithmetic if there is a number a3 d a1 The number a2 a4 d a 3 a2 such that... d. is the common difference of the arithmetic sequence |
. Example 1 Examples of Arithmetic Sequences a. The sequence whose n th term is 4n 3 is arithmetic. For this sequence, the common difference between consecutive terms is 4. 7, 11, 15, 19,..., 4n 3,... Begin with n 1. 11 7 4 b. The sequence whose n th term is 7 5n common difference between consecutive terms is 5. is arithmetic. For this sequence, the © mediacolor’s Alamy 2, 3, 8, 13,..., 7 5n,... Begin with n 1. 3 2 5 c. The sequence whose th term is n n 3 1 4 common difference between consecutive terms is,,, 1 4. Begin with n 1. is arithmetic. For this sequence, the 5 4 1 1 4 Now try Exercise 1. The sequence 1, 4, 9, 16,..., whose th term is n n2, is not arithmetic. The difference between the first two terms is 4 1 3 a1 a2 but the difference between the second and third terms is a3 a2 9 4 5. 333202_0902.qxd 12/5/05 11:29 AM Page 654 654 Chapter 9 Sequences, Series, and Probability an = dn + c an c a1 a2 a3 n FIGURE 9.3 n a1 The alternative recursion form of the th term of an arithmetic sequence can be derived from the pattern below. a1 a1 a1 a1 a1 d 2d 3d 4d a5 a4 a2 a3 4th term 5th term 2nd term 3rd term 1st term 1 less a1 n 1 d an nth term 1 less In Example 1, notice that each of the arithmetic sequences has an th term that is of the form where the common difference of the sequence is An arithmetic sequence may be thought of as a linear function whose domain is the set of natural numbers. dn c, d. n The nth Term of an Arithmetic Sequence The th term of an arithmetic sequence has the form n dn c an Linear form is the common difference between consecutive terms of the d where c a1 sequence and a1 shown in Figure 9.3. Substituting alternative recursion form for the nth term of an arithmetic sequence. A graphical representation of this definition is an yields an dn c d. d for in c an a1 n 1 |
d Alternative form Example 2 Finding the nth Term of an Arithmetic Sequence Find a formula for the difference is 3 and whose first term is 2. n th term of the arithmetic sequence whose common Solution Because the sequence is arithmetic, you know that the formula for the th term d 3, an is of the form the formula must have the form n Moreover, because the common difference is dn c. Because it follows that an 3n c. 2, a1 d c a1 2 3 1. Substitute 3 for d. Substitute 2 for a1 and 3 for d. So, the formula for the th term is n an 3n 1. The sequence therefore has the following form. 2, 5, 8, 11, 14,..., 3n 1,... Now try Exercise 21. Another way to find a formula for the th term of the sequence in Example n 2 is to begin by writing the terms of the sequence. a1 2 2 a2 2 3 5 a3 5 3 8 a4 8 3 11 a5 11 3 14 a6 14 3 17 a7 17 3 20......... From these terms, you can reason that the th term is of the form n an dn c 3n 1. 333202_0902.qxd 12/5/05 11:29 AM Page 655 in Example 3 by a1 You can find using the alternative recursion n form of the th term of an arithmetic sequence, as follows. n 1d 4 1d 4 15 15 an a1 a1 a4 20 a1 20 a1 5 a1 Section 9.2 Arithmetic Sequences and Partial Sums 655 Example 3 Writing the Terms of an Arithmetic Sequence The fourth term of an arithmetic sequence is 20, and the 13th term is 65. Write the first 11 terms of this sequence. Solution You know that d 13th terms of the sequence are related by 65. 20 and a13 a4 So, you must add the common difference nine times to the fourth term to obtain the 13th term. Therefore, the fourth and a13 9d. a4 20 a4 Using the sequence is as follows. and a13 a4 65, and a13 are nine terms apart. you can conclude that d 5, which implies that a1 5 a2 10 a3 15 a4 20 a5 25 a6 30 a7 35 a8 40 a9 45 a10 50 a11 55.. |
.... Now try Exercise 37. If you know the th term of an arithmetic sequence and you know the n th term by using the n 1 common difference of the sequence, you can find the recursion formula an1 an d. Recursion formula With this formula, you can find any term of an arithmetic sequence, provided that you know the preceding term. For instance, if you know the first term, you can find the second term. Then, knowing the second term, you can find the third term, and so on. Example 4 Using a Recursion Formula Find the ninth term of the arithmetic sequence that begins with 2 and 9. Solution There are two ways For this sequence, the common difference is to find the ninth term. One way is simply to write out the first nine terms (by repeatedly adding 7). d 9 2 7. 2, 9, 16, 23, 30, 37, 44, 51, 58 Another way to find the ninth term is to first find a formula for the Because the first term is 2, it follows that n th term. c a1 d 2 7 5. Therefore, a formula for the th term is n an 7n 5 which implies that the ninth term is 79 5 58. a9 Now try Exercise 45. 333202_0902.qxd 12/5/05 11:29 AM Page 656 656 Chapter 9 Sequences, Series, and Probability The Sum of a Finite Arithmetic Sequence There is a simple formula for the sum of a finite arithmetic sequence. Note that this formula works only for arithmetic sequences. The Sum of a Finite Arithmetic Sequence The sum of a finite arithmetic sequence with terms is n Sn n 2 a1 an. For a proof of the sum of a finite arithmetic sequence, see Proofs in Mathematics on page 723. Example 5 Finding the Sum of a Finite Arithmetic Sequence Find the sum: 1 3 5 7 9 11 13 15 17 19. Solution To begin, notice that the sequence is arithmetic (with a common difference of 2). Moreover, the sequence has 10 terms. So, the sum of the sequence is Sn a1 an Formula for the sum of an arithmetic sequence 1 19 Substitute 10 for n, 1 for a1, and 19 for an. n 2 10 Historical Note A teacher of Carl Friedrich Gauss (1777–1855) asked him to add all the integers from 1 to 100. When Gauss returned with the correct answer after only a few moments, the teacher could |
only look at him in astounded silence. This is what Gauss did:... 100 3 2 1 100 101 99 101 98 101... 1... 101 Sn Sn 2Sn Sn 100 101 2 5050 520 100. Simplify. Now try Exercise 63. Example 6 Finding the Sum of a Finite Arithmetic Sequence Find the sum of the integers (a) from 1 to 100 and (b) from 1 to N. Solution a. The integers from 1 to 100 form an arithmetic sequence that has 100 terms. So, you can use the formula for the sum of an arithmetic sequence, as follows. Sn Formula for sum of an arithmetic sequence a1 an Substitute 100 for 1 100 1 2 3 4 5 6... 99 100 n 2 100 2 50101 5050 Simplify Substitute N for 1 N an a1 1 for 1 for a1, a1, n, n, Now try Exercise 65. 100 for an. and N for an. Formula for sum of an arithmetic sequence b. Sn 333202_0902.qxd 12/5/05 11:29 AM Page 657 Section 9.2 Arithmetic Sequences and Partial Sums 657 The sum of the first n th partial sum. th partial sum can be found by using the formula for the sum of a finite terms of an infinite sequence is the n n The arithmetic sequence. Example 7 Finding a Partial Sum of an Arithmetic Sequence Find the 150th partial sum of the arithmetic sequence 5, 16, 27, 38, 49,.... Solution For this arithmetic sequence, 5 and d 16 5 11. So, a1 c a1 n th term is d 5 11 6 11n 6. and the the sum of the first 150 terms is an Therefore, a150 11150 6 1644, and S150 a150 nth partial sum formula a1 n 2 150 2 751649 123,675. 5 1644 Substitute 150 for n, 5 for a1, and 1644 for a150. Simplify. nth partial sum Now try Exercise 69. Applications Example 8 Prize Money In a golf tournament, the 16 golfers with the lowest scores win cash prizes. First place receives a cash prize of $1000, second place receives $950, third place receives $900, and so on. What is the total amount of prize money? Solution The cash prizes awarded form an arithmetic sequence in which the common difference is c a1 d 50. Because d 1000 |
50 1050 n th term of the sequence is is term sequence the of and the total amount of prize money is So, the you can determine that the formula for the an a16 50n 1050. 5016 1050 250, S16 1000 950 900... 250 n 2 a16 16th a1 S16 nth partial sum formula 16 2 1000 250 Substitute 16 for n, 1000 for a1, and 250 for a16. 81250 $10,000. Simplify. Now try Exercise 89. 333202_0902.qxd 12/5/05 11:30 AM Page 658 658 Chapter 9 Sequences, Series, and Probability Example 9 Total Sales A small business sells $10,000 worth of skin care products during its first year. The owner of the business has set a goal of increasing annual sales by $7500 each year for 9 years. Assuming that this goal is met, find the total sales during the first 10 years this business is in operation. Solution The annual sales form an arithmetic sequence in which d 7500. So, c a1 d 10,000 7500 2500 10,000 and a1 n and the th term of the sequence is 7500n 2500. an This implies that the 10th term of the sequence is a10 750010 2500 77,500. See Figure 9.4. The sum of the first 10 terms of the sequence is Small Business an a n = 7500n + 2500 S10 1 2 3 4 5 6 7 8 9 10 Year n n 2 10 2 a1 a10 nth partial sum formula 10,000 77,500 Substitute 10 for 10,000 for n, a1, and 77,500 for a10. 587,500 437,500. Simplify. Simplify. So, the total sales for the first 10 years will be $437,500. Now try Exercise 91. W RITING ABOUT MATHEMATICS Numerical Relationships Decide whether it is possible to fill in the blanks in each of the sequences such that the resulting sequence is arithmetic. If so, find a recursion formula for the sequence. a. 7, b. 17, c. 2, 6,,,, d. 4, 7.5, e. 8, 12,,,,,,,,,, 11,,,, 71, 162,,,,,,,, 39, 60.75 80,000 60,000 |
40,000 20,000 ) FIGURE 9.4 333202_0902.qxd 12/5/05 11:30 AM Page 659 Section 9.2 Arithmetic Sequences and Partial Sums 659 9.2 Exercises VOCABULARY CHECK: Fill in the blanks. 1. A sequence is called an ________ sequence if the differences between two consecutive terms are the same. This difference is called the ________ difference. 2. The th term of an arithmetic sequence has the form ________. n 3. The formula Sn n 2 a1 an can be used to find the sum of the first n terms of an arithmetic sequence, called the ________ of a ________ ________ ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–10, determine whether the sequence is arithmetic. If so, find the common difference. 10, 8, 6, 4, 2,... 2. 4, 7, 10, 13, 16,... 28. 29. 30. a1 a3 a5 16 85 4, a5 94, a6 190, a10 115 4. 80, 40, 20, 10, 5, 2, 2, 3 2, 1,... 3, 5 6.... In Exercises 31–38, write the first five terms of the arithmetic sequence. 1. 3. 5. 7. 8. 9. 10. 1, 2, 4, 8, 16,... 9 2, 5 4, 3 4, 2, 7 4,... 5 1 2 4 3, 3, 6,... 3, 1, 5.3, 5.7, 6.1, 6.5, 6.9,... ln 1, ln 2, ln 3, ln 4, ln 5,... 12, 22, 32, 42, 52,... In Exercises 11–18, write the first five terms of the sequence. Determine whether the sequence is arithmetic. If so, find the common difference. (Assume that begins with 1.) n 12. 14. an an 100 3n 1 n 14 11. 13. 15. 16. an an an an 5 3n 3 4n 2 1n 2n1 1n3 n |
2n n 17. an 18. an In Exercises 19–30, find a formula for sequence. an for the arithmetic 19. 20. 21. 22. 23. 24. 25. 26. 27. 1, d 3 15, d 4 100, d 8 0, d 2 3 x, d 2x y, d 5y 2, 1, 7 2,... a1 a1 a1 a1 a1 a1 4, 3 10, 5, 0, 5, 10,... a1 5, a4 15 31. 32. 33. 34. 35. 36. 37. 38. a1 a1 a1 a1 a1 a4 a8 a3 5, d 6 5, d 3 4 2.6, d 0.4 16.5, d 0.25 2, a12 16, a10 26, a12 19, a15 46 46 42 1.7 In Exercises 39–44, write the first five terms of the arithmetic sequence. Find the common difference and write the th term of the sequence as a function of n. n 39. 40. 41. 42. 43. 44. a1 a1 a1 a1 a1 a1 4 5 ak ak 15, ak1 6, ak1 200, ak1 ak ak 72, ak1 5 ak 8, ak1 0.375, ak1 1 8 ak 10 6 0.25 In Exercises 45–48, the first two terms of the arithmetic sequence are given. Find the missing term. 45. 46. 47. 48. a1 a1 a1 a1 5, 11, a2 3, 13, a2 4.2, a2 0.7, a7 13.8, a10 a9 6.6, a2 a8 333202_0902.qxd 12/5/05 11:30 AM Page 660 660 Chapter 9 Sequences, Series, and Probability In Exercises 49–52, match the arithmetic sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) an (b) an 24 18 12 6 − 6 (c) an 10 8 6 4 2 −2 2 4 6 8 2 4 6 8 10 8 6 4 2 −2 −4 (d) an 30 24 18 12 6 −6 n n 2 4 6 8 10 2 4 6 8 10 n n |
49. 51. an an 3 2 3 4 n 8 4 n 50. 52. an an 3n 5 25 3n 71. 73. 30 n11 n 10 n1 n 400 n1 2n 1 72. 74. 100 n51 n 50 n1 n 250 n1 1000 n In Exercises 75–80, use a graphing utility to find the partial sum. 20 1000 5n 2n 5 50 76. 75. n1 100 n1 60 i1 n 4 2 3i 250 8 77. 79. n0 78. 80. 100 n0 8 3n 16 200 j1 4.5 0.025j Job Offer the given starting salary and the given annual raise. In Exercises 81 and 82, consider a job offer with (a) Determine the salary during the sixth year of employment. (b) Determine the total compensation from the company through six full years of employment. Starting Salary Annual Raise In Exercises 53–56, use a graphing utility to graph the first 10 terms of the sequence. (Assume that begins with 1.) n 81. $32,500 82. $36,800 $1500 $1750 53. 55. an an 15 3 2n 0.2n 3 54. 56. an an 5 2n 0.3n 8 In Exercises 57– 64, find the indicated th partial sum of the arithmetic sequence. n 57. 8, 20, 32, 44,..., 58. 2, 8, 14, 20,..., n 10 n 25 59. 4.2, 3.7, 3.2, 2.7,..., 60. 0.5, 0.9, 1.3, 1.7,..., n 12 n 10 61. 40, 37, 34, 31,..., 63. 62. 75, 70, 65, 60, 100, a25 15, a100 a1 a1 64...., 220, 307, n 10 n 25 n 25 n 100 65. Find the sum of the first 100 positive odd integers. 66. Find the sum of the integers from 10 to 50. In Exercises 67–74, find the partial sum. 67. 69. 50 n1 n 100 n10 6n 68. 70. 100 n1 2n 100 n51 7n 83. Seating Capacity Determine the seating capacity of an auditorium with |
30 rows of seats if there are 20 seats in the first row, 24 seats in the second row, 28 seats in the third row, and so on. 84. Seating Capacity Determine the seating capacity of an auditorium with 36 rows of seats if there are 15 seats in the first row, 18 seats in the second row, 21 seats in the third row, and so on. 85. Brick Pattern A brick patio has the approximate shape of a trapezoid (see figure). The patio has 18 rows of bricks. The first row has 14 bricks and the 18th row has 31 bricks. How many bricks are in the patio? 31 14 FIGURE FOR 85 FIGURE FOR 86 86. Brick Pattern A triangular brick wall is made by cutting some bricks in half to use in the first column of every other row. The wall has 28 rows. The top row is one-half brick wide and the bottom row is 14 bricks wide. How many bricks are used in the finished wall? 333202_0902.qxd 12/5/05 11:30 AM Page 661 Section 9.2 Arithmetic Sequences and Partial Sums 661 87. Falling Object An object with negligible air resistance is dropped from a plane. During the first second of fall, the object falls 4.9 meters; during the second second, it falls 14.7 meters; during the third second, it falls 24.5 meters; it falls 34.3 meters. If this during the fourth second, arithmetic pattern continues, how many meters will the object fall in 10 seconds? 88. Falling Object An object with negligible air resistance is dropped from the top of the Sears Tower in Chicago at a height of 1454 feet. During the first second of fall, the object falls 16 feet; during the second second, it falls 48 feet; during the third second, it falls 80 feet; during the fourth second, it falls 112 feet. If this arithmetic pattern continues, how many feet will the object fall in 7 seconds? 89. Prize Money A county fair is holding a baked goods competition in which the top eight bakers receive cash prizes. First places receives a cash prize of $200, second place receives $175, third place receives $150, and so on. (b) Find the total amount of interest paid over the term of the loan. 94. Borrowing Money You borrowed $5000 from your parents to purchase a used car. The arrangements of the loan are such that you will make payments of $250 per month |
plus 1% interest on the unpaid balance. (a) Find the first year’s monthly payments you will make, and the unpaid balance after each month. (b) Find the total amount of interest paid over the term of the loan. Model It 95. Data Analysis: Personal Income The table shows in the United States (Source: U.S. Bureau of the per capita personal income from 1993 to 2003. Economic Analysis) an (a) Write a sequence an awarded in terms of the place good places. that represents the cash prize in which the baked n (b) Find the total amount of prize money awarded at the competition. 90. Prize Money A city bowling league is holding a tournament in which the top 12 bowlers with the highest three-game totals are awarded cash prizes. First place will win $1200, second place $1100, third place $1000, and so on. (a) Write a sequence an awarded in terms of the place finishes. that represents the cash prize in which the bowler n (b) Find the total amount of prize money awarded at the tournament. 91. Total Profit A small snowplowing company makes a profit of $8000 during its first year. The owner of the company sets a goal of increasing profit by $1500 each year for 5 years. Assuming that this goal is met, find the total profit during the first 6 years of this business. What kinds of economic factors could prevent the company from meeting its profit goal? Are there any other factors that could prevent the company from meeting its goal? Explain. 92. Total Sales An entrepreneur sells $15,000 worth of sports memorabilia during one year and sets a goal of increasing annual sales by $5000 each year for 9 years. Assuming that this goal is met, find the total sales during the first 10 years of this business. What kinds of economic factors could prevent the business from meeting its goals? 93. Borrowing Money You borrowed $2000 from a friend to purchase a new laptop computer and have agreed to pay back the loan with monthly payments of $200 plus 1% interest on the unpaid balance. (a) Find the first six monthly payments you will make, and the unpaid balance after each month. Year Per capita personal income, an 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 $21,356 $22,176 $23,078 $24,176 $25,334 $26,880 $27,933 $29,848 $30,534 $30,9 |
13 $31,633 (a) Find an arithmetic sequence that models the data. corresponding represent the year, with n 3 n Let to 1993. (b) Use the regression feature of a graphing utility to find a linear model for the data. How does this model compare with the arithmetic sequence you found in part (a)? (c) Use a graphing utility to graph the terms of the finite sequence you found in part (a). (d) Use the sequence from part (a) to estimate the per capita personal income in 2004 and 2005. (e) Use your school’s library, the Internet, or some other reference source to find the actual per capita personal income in 2004 and 2005, and compare these values with the estimates from part (d). 333202_0902.qxd 12/8/05 10:53 AM Page 662 662 Chapter 9 Sequences, Series, and Probability 96. Data Analysis: Revenue The table shows the annual (in millions of dollars) for Nextel (Source: revenue Communications, Inc. from 1997 to 2003. Nextel Communications, Inc.) an Year Revenue, an 1997 1998 1999 2000 2001 2002 2003 739 1847 3326 5714 7689 8721 10,820 (a) Construct a bar graph showing the annual revenue from 1997 to 2003. (b) Use the linear regression feature of a graphing utility to find an arithmetic sequence that approximates the annual revenue from 1997 to 2003. (c) Use summation notation to represent the total revenue from 1997 to 2003. Find the total revenue. (d) Use the sequence from part (b) to estimate the annual revenue in 2008. (d) Compare the slope of the line in part (b) with the common difference of the sequence in part (a). What can you conclude about the slope of a line and the common difference of an arithmetic sequence? 102. Pattern Recognition (a) Compute the following sums of positive odd integers 11 (b) Use the sums in part (a) to make a conjecture about the sums of positive odd integers. Check your conjecture for the sum 1 3 5 7 9 11 13. (c) Verify your conjecture algebraically. 103. Think About It The sum of the first 20 terms of an arithmetic sequence with a common difference of 3 is 650. Find the first term. terms of an arithand common difference is Determine the sum if each term is increased by 5. 104. Think About It The sum of the |
first metic sequence with first term Sn. d Explain. a1 n Synthesis Skills Review True or False? the statement is true or false. Justify your answer. In Exercises 97 and 98, determine whether In Exercises 105–108, find the slope and y-intercept (if possible) of the equation of the line. Sketch the line. 97. Given an arithmetic sequence for which only the first two terms are known, it is possible to find the th term. n 98. If the only known information about a finite arithmetic sequence is its first term and its last term, then it is possible to find the sum of the sequence. 105. 106. 107. 108. 2x 4y 3 9x y 8 x 7 0 y 11 0 99. Writing In your own words, explain what makes a sequence arithmetic. In Exercises 109 and 110, use Gauss-Jordan elimination to solve the system of equations. 100. Writing Explain how to use the first two terms of an 109. arithmetic sequence to find the th term. n 101. Exploration (a) Graph the first 10 terms of the arithmetic sequence an 2 3n. (b) Graph the equation of the line y 3x 2. (c) Discuss any differences between the graph of 110. 3x 6x 2x x 5x 8x y 2y 5y 7z 4z z 10 17 20 4y 3y 2y 10z z 3z 4 31 5 2 3n an and the graph of y 3x 2. 111. Make a Decision To work an extended application analyzing the median sales price of existing one-family homes in the United States from 1987 to 2003, visit this text’s website at college.hmco.com. (Data Source: National Association of Realtors) 333202_0903.qxd 12/5/05 11:32 AM Page 663 9.3 Geometric Sequences and Series Section 9.3 Geometric Sequences and Series 663 What you should learn • Recognize, write, and find the nth terms of geometric sequences. • Find nth partial sums of geometric sequences. • Find the sum of an infinite geometric series. • Use geometric sequences to model and solve real-life problems. Why you should learn it Geometric sequences can be used to model and solve reallife problems. For instance, in Exercise 99 on page 670, you will use a geometric sequence to model the population of China. © Bob Krist/ |
Corbis Geometric Sequences In Section 9.2, you learned that a sequence whose consecutive terms have a common difference is an arithmetic sequence. In this section, you will study another important type of sequence called a geometric sequence. Consecutive terms of a geometric sequence have a common ratio. Definition of Geometric Sequence A sequence is geometric if the ratios of consecutive terms are the same. So, r the sequence such that is geometric if there is a number a1, a2, a3, a4,..., an... a2 a1 r, a3 a2 r, a4 a3 r, r 0 and so the number r is the common ratio of the sequence. Example 1 Examples of Geometric Sequences a. The sequence whose n th term is 2n is geometric. For this sequence, the common ratio of consecutive terms is 2. 2, 4, 8, 16,..., 2n,... Begin with n 1. 4 2 2 b. The sequence whose 43n common ratio of consecutive terms is 3. 12, 36, 108, 324,..., 43n,... n th term is is geometric. For this sequence, the Begin with n 1. 36 12 3 c. The sequence whose n th term is n 1 is geometric. For this sequence, the 3 1 3. common ratio of consecutive terms is 1 81,..., 1 3, 1 27 1 3 Begin with,... n 1. n 1 9,, 19 13 1 3 Now try Exercise 1. The sequence 1, 4, 9, 16,..., whose th term is n n2, is not geometric. The ratio of the second term to the first term is a2 a1 4 1 4 but the ratio of the third term to the second term is a3 a2 9 4. 333202_0903.qxd 12/5/05 11:32 AM Page 664 664 Chapter 9 Sequences, Series, and Probability In Example 1, notice that each of the geometric sequences has an n th term that is of the form where the common ratio of the sequence is A geometric sequence may be thought of as an exponential function whose domain is the set of natural numbers. ar n, r. The nth Term of a Geometric Sequence The th term of a geometric sequence has the form n an a1r n1 r is the common ratio of consecutive terms of the |
sequence. So, every where geometric sequence can be written in the following form. a1, a2, a3, a4, a5,....., an,..... a1, a1r, a1r2, a1r3, a1r 4,..., a1rn1,... n th term of a geometric sequence, you can find the If you know the ran. term by multiplying by That is, an1 r. n 1th Example 2 Finding the Terms of a Geometric Sequence Write the first five terms of the geometric sequence whose first term is whose common ratio is and Then graph the terms on a set of coordinate axes. r 2. a1 3 an 50 40 30 20 10 1 2 3 4 5 n Solution Starting with 3, repeatedly multiply by 2 to obtain the following. a1 a2 a3 a4 a5 3 321 6 322 12 323 24 324 48 1st term 2nd term 3rd term 4th term 5th term FIGURE 9.5 Figure 9.5 shows the first five terms of this geometric sequence. Now try Exercise 11. Example 3 Finding a Term of a Geometric Sequence Find the 15th term of the geometric sequence whose first term is 20 and whose common ratio is 1.05. Solution a15 a1r n1 201.05151 39.599 Formula for geometric sequence Substitute 20 for a1, 1.05 for r, and 15 for n. Use a calculator. Now try Exercise 27. 333202_0903.qxd 12/5/05 11:32 AM Page 665 Section 9.3 Geometric Sequences and Series 665 Example 4 Finding a Term of a Geometric Sequence Find the 12th term of the geometric sequence 5, 15, 45,.... Solution The common ratio of this sequence is r 15 5 3. a1 an Because the first term is a1r n1 53121 5177,147 885,735. a12 5, you can determine the 12th term n 12 to be Formula for geometric sequence Substitute 5 for a1, 3 for r, and 12 for n. Use a calculator. Simplify. Now try Exercise 35. If you know any two terms of a geometric sequence, you can use that infor- mation to find a formula for the th term of the sequence. n Example 5 Finding a Term of a Geometric Sequence The fourth term of |
a geometric sequence is 125, and the 10th term is Find the 14th term. (Assume that the terms of the sequence are positive.) 12564. Solution The 10th term is related to the fourth term by the equation r is the common Remember that ratio of consecutive terms of a geometric sequence. So, in Example 5, a10 a1r 9 a1 r r r r 6 a1 a2 a4 a3 a1 a3 a2 a4r 6. a10 a4r 6. a10 Because 12564 Multiply 4th term by r 104. and a4 125, you can solve for as follows. r r 6 125 64 1 64 1 2 125r6 Substitute 125 64 for a10 and 125 for a4. r 6 r Divide each side by 125. Take the sixth root of each side. You can obtain the 14th term by multiplying the 10th term by r 4. a14 4 a10r 4 1 125 2 64 125 1024 Multiply the 10th term by r1410. Substitute 125 64 for a10 and 1 2 for r. Simplify. Now try Exercise 41. 333202_0903.qxd 12/5/05 11:32 AM Page 666 666 Chapter 9 Sequences, Series, and Probability The Sum of a Finite Geometric Sequence The formula for the sum of a finite geometric sequence is as follows. The Sum of a Finite Geometric Sequence The sum of the finite geometric sequence a1, a1r, a1r 2, a1r 3, a1r 4,..., a1r n1 n is given by Sn r 1 with common ratio i1 a1 r i1 a11 r n 1 r. For a proof of the sum of a finite geometric sequence, see Proofs in Mathematics on page 723. Example 6 Finding the Sum of a Finite Geometric Sequence Find the sum 12 i1 40.3i1. Solution By writing out a few terms, you have 12 i1 40.3i1 40.30 40.31 40.32... 40.311. n 12, Now, because sum of a finite geometric sequence to obtain r 0.3, 4, and a1 you can apply the formula for the Sn a11 r n 1 r 12 i1 40.3i1 41 0.312 1 0.3 5.714. Formula for the sum of a sequence Substitute 4 |
for a1, 0.3 for r, and 12 for n. Use a calculator. Now try Exercise 57. When using the formula for the sum of a finite geometric sequence, be careful to check that the sum is of the form n i1 a1 r i1. Exponent for r is i 1. If the sum is not of this form, you must adjust the formula. For instance, if the sum in Example 6 were 12 i1 40.3i, then you would evaluate the sum as follows. 12 i1 40.3i 40.3 40.32 40.33... 40.312 40.3 40.30.3 40.30.32... 40.30.311 40.31 0.312 1 0.3 40.3, r 0.3, n 12 1.714. a1 333202_0903.qxd 12/5/05 11:32 AM Page 667 Exploration Use a graphing utility to graph and What for 2, 3, 5. x →? happens as Use a graphing utility to graph y 1 r x 1 r r 1.5, for happens as x →? 2, and 3. What Section 9.3 Geometric Sequences and Series 667 Geometric Series The summation of the terms of an infinite geometric sequence is called an infinite geometric series or simply a geometric series. r, The formula for the sum of a finite geometric sequence can, depending on the value of be extended to produce a formula for the sum of an infinite geometric series. Specifically, if the common ratio has the property that it r n can be shown that increases without bound. Consequently, becomes arbitrarily close to zero as r < 1, n r a11 r n 1 r a11 0 1 r as n. This result is summarized as follows. The Sum of an Infinite Geometric Series the infinite geometric series If a1r a1r2 a1r3... a1r n1... r < 1, a1 has the sum S i0 a1r i a1 1 r. Note that if r ≥ 1, the series does not have a sum. Example 7 Finding the Sum of an Infinite Geometric Series Find each sum. 40.6n 1 n1 3 0.3 0.03 0.003... a. b. Solution a. n1 40.6n 1 4 40.6 40.62 40.63... 40 |
.6n 1... 4 1 0.6 10 a 1 1 r b. 3 0.3 0.03 0.003... 3 30.1 30.12 30.13... 3 1 0.1 a 1 1 r 10 3 3.33 Now try Exercise 79. 333202_0903.qxd 12/5/05 11:32 AM Page 668 668 Chapter 9 Sequences, Series, and Probability Application Example 8 Increasing Annuity Recall from Section 3.1 that the formula for compound interest is A P1 r n nt. A deposit of $50 is made on the first day of each month in a savings account that pays 6% compounded monthly. What is the balance at the end of 2 years? (This type of savings plan is called an increasing annuity.) Solution The first deposit will gain interest for 24 months, and its balance will be P, 0.06 is the interest So, in Example 8, $50 is the principal r, rate 12 is the number of n, and compoundings per year 2 is the time in years. If you substitute these values into the formula, you obtain t A 501 0.06 12 501 0.06 12 122 24. A24 24 501 0.06 12 501.00524. The second deposit will gain interest for 23 months, and its balance will be A23 23 501 0.06 12 501.00523. The last deposit will gain interest for only 1 month, and its balance will be A1 501 0.06 12 1 501.005. The total balance in the annuity will be the sum of the balances of the 24 deposits. Using the formula for the sum of a finite geometric sequence, with A1 501.005 r 1.005, and S24 501.0051 1.00524 1 1.005 you have 501.005 for Substitute and 24 for n. r, 1.005 for A1, $1277.96. Simplify. Now try Exercise 107. W RITING ABOUT MATHEMATICS An Experiment You will need a piece of string or yarn, a pair of scissors, and a tape measure. Measure out any length of string at least 5 feet long. Double over the string and cut it in half. Take one of the resulting halves, double it over, and cut it in half. Continue this process until you are no longer able to cut a length of string in half. How many cuts were you |
able to make? Construct a sequence of the resulting string lengths after each cut, starting with the original length of the string. Find a formula for the nth term of this sequence. How many cuts could you theoretically make? Discuss why you were not able to make that many cuts. 333202_0903.qxd 12/5/05 11:32 AM Page 669 Section 9.3 Geometric Sequences and Series 669 9.3 Exercises VOCABULARY CHECK: Fill in the blanks. 1. A sequence is called a ________ sequence if the ratios between consecutive terms are the same. This ratio is called the ________ ratio. 2. The th term of a geometric sequence has the form ________. n 3. The formula for the sum of a finite geometric sequence is given by ________. 4. The sum of the terms of an infinite geometric sequence is called a ________ ________. 5. The formula for the sum of an infinite geometric series is given by ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–10, determine whether the sequence is geometric. If so, find the common ratio. In Exercises 35– 42, find the indicated geometric sequence. n th term of the 1. 3. 5. 7. 9. 5, 15, 45, 135,..... 3, 12, 21, 30,. 1, 1 4, 1 2, 1 8,... 1 8, 1 4, 1.. 2, 1,. 3, 1 2, 1 1, 1 4,... 2. 4. 6. 8. 10. 3, 12, 48, 192,... 36, 27, 18, 9,... 5, 1, 0.2, 0.04,. 9, 6, 4, 8 3,. 1 5, 2 9, 4. 11,.. 7, 3.... In Exercises 11–20, write the first five terms of the geometric sequence. 11. 13. 15. 17. a1 a1 a1 a1 19. a1 2, r 3 1, r 1 2 5, r 1 10 1, r e 2, r x 4 12. 14. 16. 18. 6, r 2 1, r 1 3 6, r 1 4 3, r 5 a1 a1 a |
1 a1 20. a1 5, r 2x In Exercises 21–26, write the first five terms of the geometric sequence. Determine the common ratio and write the th term of the sequence as a function of n 21. 23. 25. a1 a1 a1 64, ak1 7, ak1 6, ak1 1 2ak 2a k 3 2ak 22. 24. 26. n. 81, ak1 5, ak1 48, ak1 a1 a1 a1 1 3ak 2ak 1 2 ak In Exercises 27–34, write an expression for the th term of the geometric sequence. Then find the indicated term. 2, n 8 27. 28. n 5, r 3 64, r 1 a1 a1 30. 4, n 10 2, n 10 4, r 1 6, r 1 3, n 12 100, r ex, n 9 1, r 3, n 8 500, r 1.02, n 40 1000, r 1.005, n 60 a1 a1 a1 a1 a1 a1 29. 31. 32. 33. 34. 35. 9th term: 7, 21, 63,... 36. 7th term: 3, 36, 432,... 37. 10th term: 5, 30, 180,... 40. 1st term: 39. 3rd term: 38. 22nd term: 4, 8, 16,... 27 a1 4 a2 a4 a3 16, 3, a5 18, 16 a5 3, a4 3 64 2 3 41. 6th term: 42. 7th term: a7 64 27 In Exercises 43– 46, match the geometric sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) an (b) an 20 16 12 8 4 −4 (c) an 18 12 6 −2 −12 −18 43. 45. an an 750 600 450 300 150 −2 (d) an 600 400 200 −200 −400 −600 n n 2 4 6 8 10 2 8 10 n n 2 4 6 8 10 2 8 10 182 183 3 2 n1 n1 44. an 46. an 182 183 3 n1 n1 2 333202_0903.qxd 12/5/05 11:32 AM Page 670 670 Chapter 9 Sequences |
, Series, and Probability 47. In Exercises 47–52, use a graphing utility to graph the first 10 terms of the sequence. 120.75n1 120.4n1 21.3n1 101.5n1 201.25n1 101.2n1 51. 49. 50. 48. 52. an an an an an an In Exercises 53–72, find the sum of the finite geometric sequence. 9 2 n1 n1 10 53. 54. 5 2 n1 53 2 8 n1 n1 i1 21 4 10 i1 161 2 12 i1 i1 53 5 n 40 n0 101 5 n 20 n0 6 n0 5001.04n n1 55. 57. 59. 61. 63. 65. 67. 69. 71. 9 n1 2n1 641 2 7 i1 i1 321 4 6 i1 i1 20 33 2 n0 15 24 3 n0 n n 5 n0 3001.06n 21 4 n 40 n0 81 4 10 i1 51 3 10 i1 i1 i1 56. 58. 60. 62. 64. 66. 68. 70. 72. 4 n n0 n0 41 0.4n n0 8 6 9 2 1 9 1 3 30.9n... 27 8 1 3... 83. 85. 87. 89. 91. n n0 n0 1 10 40.2n n0 9 6 4 8 3 125 25 6 36 100.2n... 5 6... 84. 86. 88. 90. 92. In Exercises 93–96, find the rational number representation of the repeating decimal. 93. 0.36 95. 0.318 94. 0.297 96. 1.38 Graphical Reasoning In Exercises 97 and 98, use a graphing utility to graph the function. Identify the horizontal asymptote of the graph and determine its relationship to the sum. 97. 98. f x 61 0.5x 1 0.5, f x 21 0.8x 1 0.8, n 0 n 0 n n 61 2 24 5 102 3 50 n0 n1 Model It 81 2 i 25 i0 152 3 100 i1 i1 99. Data Analysis: Population The table shows the of China (in millions) from 1998 through population 2004. an (Source: U.S. Census Bureau) In |
Exercises 73–78, use summation notation to write the sum. 73. 74. 75. 76. 77. 78. 5 15 45... 3645 7 14 28... 896 2 1 1 8 2 15 3 3 5... 1 2048.. 3 625. 0.1 0.4 1.6... 102.4 32 24 18... 10.125 In Exercises 79–92, find the sum of the infinite geometric series. Year 1998 1999 2000 2001 2002 2003 2004 Population, an 1250.4 1260.1 1268.9 1276.9 1284.3 1291.5 1298.8 (a) Use the exponential regression feature of a graphing utility to find a geometric sequence that models the data. Let represent the year, with n 8 corresponding to 1998. n 79. 81. n0 n0 n 1 2 1 2 n 80. 82. n0 n0 22 3 n 22 3 n (b) Use the sequence from part (a) to describe the rate at which the population of China is growing. 333202_0903.qxd 12/5/05 11:32 AM Page 671 Model It (co n t i n u e d ) (c) Use the sequence from part (a) to predict the population of China in 2010. The U.S. Census Bureau predicts the population of China will be 1374.6 million in 2010. How does this value compare with your prediction? (d) Use the sequence from part (a) to determine when the population of China will reach 1.32 billion. 100. Compound Interest A principal of $1000 is invested at 6% interest. Find the amount after 10 years if the interest (b) semiannually, (a) annually, is compounded (c) quarterly, (d) monthly, and (e) daily. 101. Compound Interest A principal of $2500 is invested at 2% interest. Find the amount after 20 years if the interest (b) semiannually, (a) annually, is compounded (c) quarterly, (d) monthly, and (e) daily. 102. Depreciation A tool and die company buys a machine for $135,000 and it depreciates at a rate of 30% per year. (In other words, at the end of each year the depreciated value is 70% of what it was at the beginning of the year |
.) Find the depreciated value of the machine after 5 full years. 103. Annuities A deposit of $100 is made at the beginning of each month in an account that pays 6%, compounded monthly. The balance in the account at the end of 5 years is A 1001 0.06 12... 1001 0.06 12 60. 1 A Find A. 104. Annuities A deposit of $50 is made at the beginning of each month in an account that pays 8%, compounded monthly. The balance in the account at the end of 5 years is A 501 0.08 12... 501 0.08 12 60. 1 A Find A. 105. Annuities A deposit of r, P dollars is made at the beginning of each month in an account earning an annual after interest rate t years is A P1 r 12 P1 r 12 compounded monthly. The balance... 2 A Show that the balance is A P1 r 12 12t 11 12 r P1 r 12 12t.. Section 9.3 Geometric Sequences and Series 671 106. Annuities A deposit of compounded continuously. The balance P dollars is made at the beginning of each month in an account earning an annual r, A interest rate A Per12 Pe 2r12... Pe12tr12. t years is after Show that the balance is A Per12er t 1. er12 1 Annuities In Exercises 107–110, consider making monthly deposits of dollars in a savings account earning an annual P interest rate Use the results of Exercises 105 and 106 to find the balance years if the interest is A compounded (a) monthly and (b) continuously. after r. t 107. 108. 109. 110. P $50, r 7%, t 20 years P $75, r 3%, t 25 years P $100, r 10%, t 40 years P $20, r 6%, t 50 years P r, 111. Annuities Consider an initial deposit of dollars in an compounded account earning an annual interest rate W monthly. At the end of each month, a withdrawal of t dollars will occur and the account will be depleted in years. The amount of the initial deposit required is 2 W1 r P W1 r 12 12 W1 r 12... 12t 1. Show that the initial deposit is 12t. P W12 r 1 1 r 12 112. Annuities Determine the amount required in a retirement |
account for an individual who retires at age 65 and wants an income of $2000 from the account each month for 20 years. Use the result of Exercise 111 and assume that the account earns 9% compounded monthly. Multiplier Effect In Exercises 113–116, use the following information. A tax rebate has been given to property owners by the state government with the anticipation that of the each property owner spends approximately rebate, and in turn each recipient of this amount spends p% of what they receive, and so on. Economists refer to this exchange of money and its circulation within the economy as the “multiplier effect.” The multiplier effect operates on the idea that the expenditures of one individual become the income of another individual. For the given tax rebate, find the total amount put back into the state’s economy, if this effect continues without end. p% Tax rebate 113. $400 114. $250 115. $600 116. $450 p% 75% 80% 72.5% 77.5% 333202_0903.qxd 12/5/05 11:32 AM Page 672 672 Chapter 9 Sequences, Series, and Probability 117. Geometry The sides of a square are 16 inches in length. A new square is formed by connecting the midpoints of the sides of the original square, and two of the resulting triangles are shaded (see figure). If this process is repeated five more times, determine the total area of the shaded region. 118. Sales The annual sales an (in millions of dollars) for Urban Outfitters for 1994 through 2003 can be approximated by the model an 54.6e0.172n, n 4, 5,..., 13 n 4 n represents the year, with where corresponding to 1994. Use this model and the formula for the sum of a finite geometric sequence to approximate the total sales earned during this 10-year period. (Source: Urban Outfitters Inc.) 119. Salary An investment firm has a job opening with a salary of $30,000 for the first year. Suppose that during the next 39 years, there is a 5% raise each year. Find the total compensation over the 40-year period. 120. Distance A ball is dropped from a height of 16 feet. Each time it drops h feet, it rebounds 0.81h feet. (a) Find the total vertical distance traveled by the ball. (b) The ball takes the following times (in seconds |
) for each fall. s1 s2 s3 s4 16t 2 16, 16t 2 160.81, 16t 2 160.812, 16t 2 160.813,... 16t 2 160.81n1, sn 0 if t 1 0 if t 0.9 0 if t 0.9 2 0 if t 0.93... 0 if t 0.9n1 s1 s2 s3 s4 sn s2, the ball takes the same amount of Beginning with time to bounce up as it does to fall, and so the total time elapsed before it comes to rest is t 1 2 0.9n. n 1 122. You can find the n th term of a geometric sequence by multiplying its common ratio by the first term of the sequence raised to the n 1 th power. 123. Writing Write a brief paragraph explaining why the terms of a geometric sequence decrease in magnitude when 1 < r < 1. 124. Find two different geometric series with sums of 4. Skills Review the function for 125. and gx x 2 1. In Exercises 125–128, evaluate f x 3x 1 gx 1 f x 1 f gx 1 g f x 1 126. 127. 128. In Exercises 129–132, completely factor the expression. 129. 130. 131. 132. 9x3 64x x2 4x 63 6x2 13x 5 16x2 4x 4 In Exercises 133–138, perform the indicated operation(s) and simplify. xx 3 x 3 2xx 7 6xx 2 6x 3 10 2x 23 x 2 x 2 133. 134. 135. 136. 137. 138 3x 1x 4 Find this total time. Synthesis True or False? In Exercises 121 and 122, determine whether the statement is true or false. Justify your answer. 121. A sequence is geometric if the ratios of consecutive differences of consecutive terms are the same. 139. Make a Decision To work an extended application analyzing the amounts spent on research and development in the United States from 1980 to 2003, visit this text’s website at college.hmco.com. (Data Source: U.S. Census Bureau) 333202_0904.qxd 12/5/05 11:35 AM Page 673 9.4 Mathematical Induction Section 9.4 Mathematical Induction 673 What you should learn • Use mathematical induction to prove statements involving a positive |
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