Matrices and Determinants

Transcription

1 Matrices and Determinants Matrices and Systems of Equations Operations with Matrices The Inverse of a Square Matri The Determinant of a Square Matri Applications of Matrices and Determinants In Mathematics Matrices are used to model and solve a variety of problems For instance, you can use matrices to solve systems of linear equations In Real Life Matrices are used to model inventory levels, electrical networks, investment portfolios, and other real-life situations For instance, you can use a matri to model the number of people in the United States who participate in snowboarding (See Eercise, page ) Graham Heywood/istockphotocom IN CAREERS There are many careers that use matrices Several are listed below Bank Teller Eercise, page Political Analyst Eercise 7, page 97 Small Business Owner Eercises 9 7, pages and 7 Florist Eercise 7, page 7 9

2 7 Chapter Matrices and Determinants MATRICES AND SYSTEMS OF EQUATIONS What you should learn Write matrices and identify their orders Perform elementary row operations on matrices Use matrices and Gaussian elimination to solve systems of linear equations Use matrices and Gauss-Jordan elimination to solve systems of linear equations Why you should learn it You can use matrices to solve systems of linear equations in two or more variables For instance, in Eercise on page, you will use a matri to find a model for the path of a ball thrown by a baseball player Matrices In this section, you will study a streamlined technique for solving systems of linear equations This technique involves the use of a rectangular array of real numbers called a matri The plural of matri is matrices Definition of Matri If m and n are positive integers, an m n (read m by n ) matri is a rectangular array Row Row Row Row m Column Column Column Column n a a a a n a a a a n a a a a ṇ a m a m a m a mn in which each entry, a i j, of the matri is a number An m n matri has m rows and n columns Matrices are usually denoted by capital letters The entry in the ith row and jth column is denoted by the double subscript notation a ij For instance, a refers to the entry in the second row, third column A matri having m rows and n columns is said to be of order m n If m n, the matri is square of order m m or n n For a square matri, the entries a, a, a, are the main diagonal entries Eample Order of Matrices Foto Agency/PhotoLibrary Determine the order of each matri a b c d Solution 7 a This matri has one row and one column The order of the matri is b This matri has one row and four columns The order of the matri is c This matri has two rows and two columns The order of the matri is d This matri has three rows and two columns The order of the matri is Now try Eercise 9 A matri that has only one row is called a row matri, and a matri that has only one column is called a column matri

3 Section Matrices and Systems of Equations 7 The vertical dots in an augmented matri separate the coefficients of the linear system from the constant terms A matri derived from a system of linear equations (each written in standard form with the constant term on the right) is the augmented matri of the system Moreover, the matri derived from the coefficients of the system (but not including the constant terms) is the coefficient matri of the system System: Augmented Matri: Coefficient Matri: y y z z z Note the use of for the missing coefficient of the y-variable in the third equation, and also note the fourth column of constant terms in the augmented matri When forming either the coefficient matri or the augmented matri of a system, you should begin by vertically aligning the variables in the equations and using zeros for the coefficients of the missing variables Eample Writing an Augmented Matri Write the augmented matri for the system of linear equations y z w What is the order of the augmented matri? Solution y w z w y z Begin by rewriting the linear system and aligning the variables y w 9 y z w z w y z 9 Net, use the coefficients and constant terms as the matri entries Include zeros for the coefficients of the missing variables R 9 R R R The augmented matri has four rows and five columns, so it is a matri The notation R n is used to designate each row in the matri For eample, Row is represented by R Now try Eercise 7

4 7 Chapter Matrices and Determinants Elementary Row Operations In Section 7, you studied three operations that can be used on a system of linear equations to produce an equivalent system Interchange two equations Multiply an equation by a nonzero constant Add a multiple of an equation to another equation In matri terminology, these three operations correspond to elementary row operations An elementary row operation on an augmented matri of a given system of linear equations produces a new augmented matri corresponding to a new (but equivalent) system of linear equations Two matrices are row-equivalent if one can be obtained from the other by a sequence of elementary row operations Elementary Row Operations Interchange two rows Multiply a row by a nonzero constant Add a multiple of a row to another row Although elementary row operations are simple to perform, they involve a lot of arithmetic Because it is easy to make a mistake, you should get in the habit of noting the elementary row operations performed in each step so that you can go back and check your work Eample Elementary Row Operations TECHNOLOGY Most graphing utilities can perform elementary row operations on matrices Consult the user s guide for your graphing utility for specific keystrokes After performing a row operation, the new row-equivalent matri that is displayed on your graphing utility is stored in the answer variable You should use the answer variable and not the original matri for subsequent row operations a Interchange the first and second rows of the original matri Original Matri b Multiply the first row of the original matri by Original Matri c Add times the first row of the original matri to the third row Original Matri New Row-Equivalent Matri R R Note that the elementary row operation is written beside the row that is changed Now try Eercise 7 New Row-Equivalent Matri R R New Row-Equivalent Matri R

5 Section Matrices and Systems of Equations 7 In Eample in Section 7, you used Gaussian elimination with back-substitution to solve a system of linear equations The net eample demonstrates the matri version of Gaussian elimination The two methods are essentially the same The basic difference is that with matrices you do not need to keep writing the variables Eample Comparing Linear Systems and Matri Operations WARNING / CAUTION Arithmetic errors are often made when elementary row operations are performed Note the operation you perform in each step so that you can go back and check your work Remember that you should check a solution by substituting the values of, y, and z into each equation of the original system For eample, you can check the solution to Eample as follows Equation : 9 Equation : Equation : 7 Linear System y z 9 y y z 7 Add the first equation to the second equation y z 9 y z y z 7 Associated Augmented Matri 9 7 Add the first row to the second row R R Add times the first equation Add times the first row to the third equation to the third row R R 9 y z 9 y z y z Add the second equation to the third equation y z 9 y z z Multiply the third equation by y z 9 y z z At this point, you can use back-substitution to find and y y y 9 Substitute for z Solve for y Multiply the third row by Substitute for y and for z Solve for The solution is, y, and z Now try Eercise 9 R R R R R R R 9 7 Add the second row to the third row R R 9 9 R

6 7 Chapter Matrices and Determinants The last matri in Eample is said to be in row-echelon form The term echelon refers to the stair-step pattern formed by the nonzero elements of the matri To be in this form, a matri must have the following properties Row-Echelon Form and Reduced Row-Echelon Form A matri in row-echelon form has the following properties Any rows consisting entirely of zeros occur at the bottom of the matri For each row that does not consist entirely of zeros, the first nonzero entry is (called a leading ) For two successive (nonzero) rows, the leading in the higher row is farther to the left than the leading in the lower row A matri in row-echelon form is in reduced row-echelon form if every column that has a leading has zeros in every position above and below its leading It is worth noting that the row-echelon form of a matri is not unique That is, two different sequences of elementary row operations may yield different row-echelon forms However, the reduced row-echelon form of a given matri is unique Eample Row-Echelon Form Determine whether each matri is in row-echelon form If it is, determine whether the matri is in reduced row-echelon form a b c d e f Solution The matrices in (a), (c), (d), and (f) are in row-echelon form The matrices in (d) and (f) are in reduced row-echelon form because every column that has a leading has zeros in every position above and below its leading The matri in (b) is not in row-echelon form because a row of all zeros does not occur at the bottom of the matri The matri in (e) is not in row-echelon form because the first nonzero entry in Row is not a leading Now try Eercise Every matri is row-equivalent to a matri in row-echelon form For instance, in Eample, you can change the matri in part (e) to row-echelon form by multiplying its second row by

7 Section Matrices and Systems of Equations 7 Gaussian Elimination with Back-Substitution Gaussian elimination with back-substitution works well for solving systems of linear equations by hand or with a computer For this algorithm, the order in which the elementary row operations are performed is important You should operate from left to right by columns, using elementary row operations to obtain zeros in all entries directly below the leading s Eample Gaussian Elimination with Back-Substitution Solve the system y z y z y z y 7z Solution R R 7 9 R R R R R R 9 R R The matri is now in row-echelon form, and the corresponding system is y z y z z w w w 7 Using back-substitution, you can determine that the solution is, y, z, and w Now try Eercise w w w 9 9 Write augmented matri Interchange R and R so first column has leading in upper left corner Perform operations on R and R so first column has zeros below its leading Perform operations on R so second column has zeros below its leading Perform operations on R and R so third and fourth columns have leading s

8 7 Chapter Matrices and Determinants The procedure for using Gaussian elimination with back-substitution is summarized below Gaussian Elimination with Back-Substitution Write the augmented matri of the system of linear equations Use elementary row operations to rewrite the augmented matri in row-echelon form Write the system of linear equations corresponding to the matri 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 of all zeros ecept 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 Eample 7 A System with No Solution Solve the system Solution R R R R R R Write augmented matri Perform row operations Perform row operations Note that the third row of this matri consists entirely of zeros ecept 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 R R y z y z y 7z y z z y z y z 7 7 Because the third equation is not possible, the system has no solution Now try Eercise

9 Section Matrices and Systems of Equations 77 Gauss-Jordan Elimination With Gaussian elimination, elementary row operations are applied to a matri to obtain a (row-equivalent) row-echelon form of the matri A second method of elimination, called Gauss-Jordan elimination, after Carl Friedrich Gauss and Wilhelm Jordan ( 99), continues the reduction process until a reduced row-echelon form is obtained This procedure is demonstrated in Eample Eample Gauss-Jordan Elimination TECHNOLOGY For a demonstration of a graphical approach to Gauss-Jordan elimination on a matri, see the Visualizing Row Operations Program available for several models of graphing calculators at the website for this tet at academiccengagecom The advantage of using Gauss- Jordan 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 Eample Use Gauss-Jordan elimination to solve the system Solution In Eample, Gaussian elimination was used to obtain the row-echelon form of the linear system above 9 Now, apply elementary row operations until you obtain zeros above each of the leading s, as follows R R Perform operations on R so second column has a 9 zero above its leading 9R R R R The matri is now in reduced row-echelon form Converting back to a system of linear equations, you have y z Now you can simply read the solution,, y, and z, which can be written as the ordered triple,, Now try Eercise 7 9 y y y z z 9 7 Perform operations on R and R so third column has zeros above its leading The elimination procedures described in this section sometimes result in fractional coefficients For instance, in the elimination procedure for the system y y y z z 7 you may be inclined to multiply the first row by to produce a leading, 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

10 7 Chapter 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 Eample 9 A System with an Infinite Number of Solutions In Eample 9, and y are solved for in terms of the third variable z To write a solution of the system that does not use any of the three variables of the system, let a represent any real number and let z a Then solve for and y The solution can then be written in terms of a, which is not one of the variables of the system Solve the system y z y Solution R R R R The corresponding system of equations is z y z Solving for and y in terms of z, you have z and To write a solution of the system that does not use any of the three variables of the system, let a represent any real number and let z a R R Now substitute a for z in the equations for and y z a y z a So, the solution set can be written as an ordered triple with the form a, a, a where a is any real number Remember that a solution set of this form represents an infinite number of solutions Try substituting values for a to obtain a few solutions Then check each solution in the original system of equations Now try Eercise 79 y z

11 Section Matrices and Systems of Equations 79 EXERCISES VOCABULARY: Fill in the blanks See wwwcalcchatcom for worked-out solutions to odd-numbered eercises A rectangular array of real numbers that can be used to solve a system of linear equations is called a A matri is if the number of rows equals the number of columns For a square matri, the entries a, a, a,, a nn are the entries A matri with only one row is called a matri, and a matri with only one column is called a matri The matri derived from a system of linear equations is called the matri of the system The matri derived from the coefficients of a system of linear equations is called the matri 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 A matri in row-echelon form is in if every column that has a leading has zeros in every position above and below its leading SKILLS AND APPLICATIONS In Eercises 9, determine the order of the matri In Eercises, write the augmented matri for the system of linear equations y y 7 y z y z y 9 7 y z 9 z In Eercises, write the system of linear equations represented by the augmented matri (Use variables, y, z, and w, if applicable) y 9y 7 9 y z y z y z z y z In Eercises 7, fill in the blank(s) using elementary row operations to form a row-equivalent matri

12 Chapter Matrices and Determinants In Eercises, identify the elementary row operation(s) being performed to obtain the new row-equivalent matri Original Matri 7 Original Matri Original Matri New Row-Equivalent Matri 9 New Row-Equivalent Matri 9 Perform the sequence of row operations on the matri What did the operations accomplish? (a) Add times to R (b) Add times to R (c) Add times to R (d) Multiply by (e) Add times to R Perform the sequence of row operations on the matri What did the operations accomplish? 7 (a) Add Original Matri 7 R R to R (b) Interchange 7 7 R 7 R R R R and R New Row-Equivalent Matri New Row-Equivalent Matri (c) Add times to R (d) Add 7 times to R (e) Multiply (f) Add the appropriate multiples of to R,R, and R In Eercises, determine whether the matri is in row-echelon form If it is, determine if it is also in reduced row-echelon form In Eercises, write the matri in row-echelon form (Remember that the row-echelon form of a matri is not unique) 7 7 In Eercises 9, use the matri capabilities of a graphing utility to write the matri in reduced row-echelon form 9 R R R by In Eercises, write the system of linear equations represented by the augmented matri Then use backsubstitution to solve (Use variables, y, and z, if applicable) 9 R 7 9 7

13 Section Matrices and Systems of Equations 7 In Eercises 9, an augmented matri that represents a system of linear equations (in variables, y, and z, if applicable) has been reduced using Gauss-Jordan elimination Write the solution represented by the augmented matri 9 In Eercises, use matrices to solve the system of equations (if possible) Use Gaussian elimination with back-substitution or Gauss-Jordan elimination y 7 y 7 7 y 9 y 7 7 y 7 7 z y z y z 7 7 y z y z y z y z y z y z 77 7 y y 79 y z 7y z y y y y y y y y y y 9 z z z z w w w w y y 7 y y y y y y y y 7 y y z y z y y y y y y z z z z z 7 y z y z 9 y y y 9 In Eercises 9, use the matri capabilities of a graphing utility to reduce the augmented matri corresponding to the system of equations, and solve the system y z y z y z y z In Eercises 9 9, determine whether the two systems of linear equations yield the same solution If so, find the solution using matrices 9 (a) (b) y z y z y z y z z z 9 (a) (b) y z y z z 9 (a) (b) y z 7 y 7z z 9 (a) (b) y z 9 y z z In Eercises 9 9, use a system of equations to find the quadratic function f a b c that satisfies the equations Solve the system using matrices 9 y y y y y y y y y y y y y y z w w z w z w y z w y z w y z w y z w z z z z z z z w w w w w w z w w z w 9 f, f, f 9 f, f 9, f 9 y z y z y z y z y y z z y z y z z y z y z z 9

14 Chapter Matrices and Determinants 97 9 f, f 7, f f, f, f In Eercises 99, use a system of equations to find the cubic function f a b c d that satisfies the equations Solve the system using matrices 99 f f f f f f f f 7 Use the system y z y z y z to write two different matrices in row-echelon form that yield the same solution ELECTRICAL NETWORK The currents in an electrical network are given by the solution of the system I I I I I I I where I, I, and I are measured in amperes Solve the system of equations using matrices PARTIAL FRACTIONS Use a system of equations to write the partial fraction decomposition of the rational epression Solve the system using matrices PARTIAL FRACTIONS Use a system of equations to write the partial fraction decomposition of the rational epression Solve the system using matrices A A f f f f 7 f 7 f f f 7 B B C C 7 FINANCE A small shoe corporation borrowed $,, to epand its line of shoes Some of the money was borrowed at 7%, some at %, and some at % Use a system of equations to determine how much was borrowed at each rate if the annual interest was $, and the amount borrowed at % was times the amount borrowed at 7% Solve the system using matrices FINANCE A small software corporation borrowed $, to epand its software line Some of the money was borrowed at 9%, some at %, and some at % Use a system of equations to determine how much was borrowed at each rate if the annual interest was $, and the amount borrowed at % was times the amount borrowed at 9% Solve the system using matrices 9 TIPS A food server eamines the amount of money earned in tips after working an -hour shift The server has a total of $9 in denominations of $, $, $, and $ bills The total number of paper bills is The number of $ bills is times the number of $ bills, and the number of $ bills is less than twice the number of $ bills Write a system of linear equations to represent the situation Then use matrices to find the number of each denomination BANKING A bank teller is counting the total amount of money in a cash drawer at the end of a shift There is a total of $ in denominations of $, $, $, and $ bills The total number of paper bills is The number of $ bills is twice the number of $ bills, and the number of $ bills is more than the number of $ bills Write a system of linear equations to represent the situation Then use matrices to find the number of each denomination In Eercises and, use a system of equations to find the equation of the parabola y a b c that passes through the points Solve the system using matrices Use a graphing utility to verify your results y (, ) (, ) (, ) (, 9) (, ) (, ) MATHEMATICAL MODELING A video 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 coordinates obtained are shown in the table ( and y are measured in feet) Horizontal distance, y Height, y 9

15 Section Matrices and Systems of Equations (a) Use a system of equations to find the equation of the parabola y a b c that passes through the three points Solve the system using matrices (b) Use a graphing utility to graph the parabola (c) Graphically approimate the maimum height of the ball and the point at which the ball struck the ground (d) Analytically find the maimum height of the ball and the point at which the ball struck the ground (e) Compare your results from parts (c) and (d) DATA ANALYSIS: SNOWBOARDERS The table shows the numbers of people y (in millions) in the United States who participated in snowboarding in selected years from to 7 (Source: National Sporting Goods Association) Year 7 (a) Use a system of equations to find the equation of the parabola y at bt c that passes through the points Let t represent the year, with t corresponding to Solve the system using matrices (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 9 Does your answer seem reasonable? Eplain (d) Do you believe that the equation can be used for years far beyond 7? Eplain NETWORK ANALYSIS In Eercises and, 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) Water flowing through a network of pipes (in thousands of cubic meters per hour) is shown in the figure Number, y 7 (a) Solve this system using matrices for the water flow represented by i, i,,, 7 (b) Find the network flow pattern when and 7 (c) Find the network flow pattern when and The flow of traffic (in vehicles per hour) through a network of streets is shown in the figure (a) Solve this system using matrices for the traffic flow represented by i, i,,, (b) Find the traffic flow when and (c) Find the traffic flow when and EXPLORATION TRUE OR FALSE? In Eercises 7 and, determine whether the statement is true or false Justify your answer 7 7 is a matri The method of Gaussian elimination reduces a matri until a reduced row-echelon form is obtained 9 THINK ABOUT IT The augmented matri below represents system of linear equations (in variables, y, and z) that has been reduced using Gauss-Jordan elimination Write a system of equations with nonzero coefficients that is represented by the reduced matri (There are many correct answers) THINK ABOUT IT (a) Describe the row-echelon form of an augmented matri that corresponds to a system of linear equations that is inconsistent (b) Describe the row-echelon form of an augmented matri that corresponds to a system of linear equations that has an infinite number of solutions Describe the three elementary row operations that can be performed on an augmented matri What is the relationship between the three elementary row operations performed on an augmented matri and the operations that lead to equivalent systems of equations? CAPSTONE In your own words, describe the difference between a matri in row-echelon form and a matri in reduced row-echelon form Include an eample of each to support your eplanation

16 Chapter Matrices and Determinants OPERATIONS WITH MATRICES What you should learn Decide whether two matrices are equal Add and subtract matrices and multiply matrices by scalars Multiply two matrices Use matri operations to model and solve real-life problems Why you should learn it Matri operations can be used to model and solve real-life problems For instance, in Eercise 7 on page 9, matri operations are used to analyze annual health care costs Equality of Matrices In Section, 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 net two introduce some fundamentals of matri theory It is standard mathematical convention to represent matrices in any of the following three ways Representation of Matrices A matri can be denoted by an uppercase letter such as A, B, or C A matri can be denoted by a representative element enclosed in brackets, such as a ij, b ij, or c ij A matri can be denoted by a rectangular array of numbers such as A a ij a a a a m a a a a m a a a a m a n a n a ṇ a mn Two matrices A a ij and B b ij are equal if they have the same order m n and a ij b ij for i m and j n In other words, two matrices are equal if their corresponding entries are equal Royalty-Free/Corbis Eample Equality of Matrices Solve for a, a, a, and in the following matri equation a a a a Solution Because two matrices are equal only if their corresponding entries are equal, you can conclude that a, a, a a, Now try Eercise 7 and a 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, but

17 Section Operations with Matrices Matri Addition and Scalar Multiplication In this section, three basic matri operations will be covered The first two are matri addition and scalar multiplication With matri addition, you can add two matrices (of the same order) by adding their corresponding entries HISTORICAL NOTE The Granger Collection Arthur Cayley ( 9), a British mathematician, invented matrices around 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 (9 ) and his son Charles S Peirce (9 9), are credited with developing matri algebra a b c Definition of Matri Addition If A a ij and B b ij are matrices of order m n, their sum is the m n matri given by A B a ij b ij The sum of two matrices of different orders is undefined Eample d The sum of A B Addition of Matrices and is undefined because A is of order and B is of order Now try Eercise (a) In operations with matrices, numbers are usually referred to as scalars In this tet, scalars will always be real numbers You can multiply a matri A by a scalar c by multiplying each entry in A by c Definition of Scalar Multiplication If A a ij is an m n matri and c is a scalar, the scalar multiple of A by c is the m n matri given by ca ca ij

18 Chapter Matrices and Determinants The symbol A represents the negation of A, which is the scalar product A Moreover, if A and B are of the same order, then A B represents the sum of A and B That is, A B A B Subtraction of matrices The order of operations for matri epressions is similar to that for real numbers In particular, you perform scalar multiplication before matri addition and subtraction, as shown in Eample (c) Eample Scalar Multiplication and Matri Subtraction For the following matrices, find (a) A, (b) B, and (c) A B A and B Solution a A 9 Scalar multiplication Multiply each entry by Simplify b B Definition of negation Multiply each entry by c A B 9 Matri subtraction 7 Subtract corresponding entries Now try Eercise (b), (c), and (d) It is often convenient to rewrite the scalar multiple ca by factoring c out of every entry in the matri For instance, in the following eample, the scalar has been factored out of the matri

19 Section Operations with Matrices 7 The properties of matri addition and scalar multiplication are similar to those of addition and multiplication of real numbers You can review the properties of addition and multiplication of real numbers (and other properties of real numbers) in Appendi A Properties of Matri Addition and Scalar Multiplication Let A, B, and C be m n matrices and let c and d be scalars A B B A Commutative Property of Matri Addition Associative Property of Matri Addition A B C A B C cda cda) Associative Property of Scalar Multiplication A A Scalar Identity Property ca B ca cb Distributive Property c da ca da Distributive Property Note that the Associative Property of Matri Addition allows you to write epressions such as A B C 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 Eample Addition of More than Two Matrices By adding corresponding entries, you obtain the following sum of four matrices Now try Eercise 9 Eample Using the Distributive Property TECHNOLOGY Most graphing utilities have the capability of performing matri 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 and A B Perform the indicated matri operations Solution Now try Eercise In Eample, you could add the two matrices first and then multiply the matri by, as follows Notice that you obtain the same result

20 Chapter Matrices and Determinants One important property of addition of real numbers is that the number is the additive identity That is, c c for any real number c For matrices, a similar property holds That is, if A is an m n matri and O is the m n zero matri consisting entirely of zeros, then A O A In other words, O is the additive identity for the set of all m n matrices For eample, the following matrices are the additive identities for the sets of all and matrices O and O WARNING / CAUTION Remember that matrices are denoted by capital letters So, when you solve for X, you are solving for a matri that makes the matri equation true zero matri zero matri The algebra of real numbers and the algebra of matrices have many similarities For eample, compare the following solutions Real Numbers (Solve for ) a b a a b a b a b a m n Matrices (Solve for X) X A B X A A B A X O B A X B A The algebra of real numbers and the algebra of matrices also have important differences, which will be discussed later Eample Solving a Matri Equation Solve for X in the equation X A B, where A Solution and Begin by solving the matri equation for X to obtain X B A X B A B Now, using the matrices A and B, you have X Now try Eercise Substitute the matrices Subtract matri A from matri B Multiply the matri by

21 Section Operations with Matrices 9 Matri Multiplication Another basic matri operation is matri 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 Matri Multiplication If A a ij is an m n matri and B b ij is an n p matri, the product AB is an m p matri AB c ij where c i j a i b j a i b j a i b j a in b nj The definition of matri multiplication indicates a row-by-column multiplication, where the entry in the ith row and jth column of the product AB is obtained by multiplying the entries in the ith row of A by the corresponding entries in the jth column of B and then adding the results So for the product of two matrices to be defined, the number of columns of the first matri must equal the number of rows of the second matri The general pattern for matri multiplication is as follows a a a a i a m a a a a i a m a a a a i a m a n a n a n a in a mn b b b b n b b b b n b j b j b j b nj b p b p b p b np c c c i c m c c c i c m c j c j c ij c mj c p c p c ip c mp a i b j a i b j a i b j a in b nj c ij In Eample 7, the product AB is defined because the number of columns of A is equal to the number of rows of B Also, note that the product AB has order Eample 7 Finding the Product of Two Matrices Find the product using A and AB Solution To find the entries of the product, multiply each row of A by each column of B AB 9 Now try Eercise B

22 9 Chapter Matrices and Determinants Be sure you understand that for the product of two matrices to be defined, the number of columns of the first matri must equal the number of rows of the second matri 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 Equal Order of AB B n p AB m p Eample Finding the Product of Two Matrices Find the product AB where A Solution 7 and Now try Eercise B Note that the order of A is and the order of B is So, the product AB has order AB Eample 9 Patterns in Matri Multiplication a b c The product AB for the following matrices is not defined A 9 and Now try Eercise 9 B

23 Section Operations with Matrices 9 a Eample Patterns in Matri Multiplication Now try Eercise b In Eample, note that the two products are different Even if both AB and BA are defined, matri multiplication is not, in general, commutative That is, for most matrices, AB BA This is one way in which the algebra of real numbers and the algebra of matrices differ Properties of Matri Multiplication Let A, B, and C be matrices and let c be a scalar ABC ABC Associative Property of Matri Multiplication Distributive Property AB C AB AC A B)C AC BC Distributive Property cab cab AcB Associative Property of Scalar Multiplication Definition of Identity Matri The n n matri that consists of s on its main diagonal and s elsewhere is called the identity matri of order n n and is denoted by I n Identity matri Note that an identity matri must be square When the order is understood to be n n, you can denote simply by I I n If A is an n n matri, the identity matri has the property that AI n A and I n A A For eample, and AI A IA A

24 9 Chapter Matrices and Determinants Applications Matri multiplication can be used to represent a system of linear equations Note how the system The column matri B is also called a constant matri Its entries are the constant terms in the system of equations a a a b a a a b a a a b can be written as the matri equation AX B, where A is the coefficient matri of the system, and X and B are column matrices a a a a a a a a a b b b A X B Eample Solving a System of Linear Equations The notation A B represents the augmented matri formed when matri B is adjoined to matri The notation I A X represents the reduced row-echelon form of the augmented matri that yields the solution of the system Consider the following system of linear equations a Write this system as a matri equation, AX B b Use Gauss-Jordan elimination on the augmented matri A B to solve for the matri X Solution a In matri form, AX B, the system can be written as follows b The augmented matri is formed by adjoining matri B to matri A A B Using Gauss-Jordan elimination, you can rewrite this equation as I X So, the solution of the system of linear equations is,, and, and the solution of the matri equation is X Now try Eercise

25 Section Eample Operations with Matrices 9 Softball Team Epenses Two softball teams submit equipment lists to their sponsors Bats Women s Team Men s Team Balls Gloves 7 Each bat costs $, each ball costs $, and each glove costs $ 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 matri form as Notice in Eample that you cannot find the total cost using the product EC because EC is not defined That is, the number of columns of E ( columns) does not equal the number of rows of C ( row) E 7 C and The total cost of equipment for each team is given by the product CE 7 7 So, the total cost of equipment for the women s team is $ and the total cost of equipment for the men s team is $ Now try Eercise 9 CLASSROOM DISCUSSION Problem Posing Write a matri multiplication application problem that uses the matri A! [7 ] Echange problems with another student in your class Form the matrices that represent the problem, and solve the problem Interpret your solution in the contet 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

26 9 Chapter Matrices and Determinants VOCABULARY EXERCISES In Eercises, fill in the blanks Two matrices are if all of their corresponding entries are equal When performing matri operations, real numbers are often referred to as See wwwcalcchatcom for worked-out solutions to odd-numbered eercises A matri consisting entirely of zeros is called a matri and is denoted by The n n matri consisting of s on its main diagonal and s elsewhere is called the matri of order n n In Eercises and, match the matri property with the correct form A, B, and C are matrices of order m n, and c and d are scalars (a) A A (i) Distributive Property (b) (c) (d) (e) (ii) Commutative Property of Matri Addition (iii) Scalar Identity Property (iv) Associative Property of Matri Addition (v) Associative Property of Scalar Multiplication (a) A O A (i) Distributive Property (b) cab AcB (ii) Additive Identity of Matri Addition (c) (d) A B C A B C c da ca da cda cda A B B A AB C AB AC ABC ABC SKILLS AND APPLICATIONS (iii) Associative Property of Matri Multiplication (iv) Associative Property of Scalar Multiplication In Eercises 7, find and y 7 7 y 7 9 In Eercises, if possible, find (a) A B, (b) A B, (c) A, and (d) A B 7 A A A A y,,, 9, B B B B y y 7 y 7 7 In Eercises 9, evaluate the epression 9 A B A A A, 7,, B, 7 B B 9 7 7

27 Section Operations with Matrices 9 In Eercises, use the matri capabilities of a graphing utility to evaluate the epression Round your results to three decimal places, if necessary 7 In Eercises 9, solve for X in the equation, given [ [ A and B ] ] In Eercises, if possible, find AB and state the order of the result A A 7 A A A, B, B, B 9 7, B 7, B X A B X A B 9 X A B A B X 9 In Eercises, use the matri capabilities of a graphing utility to find AB, if possible 7 A 7, B In Eercises 7, if possible, find (a) AB, (b) BA, and (c) A (Note: A AA ) 7 9 A A, A A A A A 9 B A 9 A A A,, A, 7 A,, B B 9, B 9, B 7 7,, B 7 7 B, B B, 7, B 7 B A, B B B

28 9 Chapter Matrices and Determinants In Eercises, evaluate the epression Use the matri capabilities of a graphing utility to verify your answer MANUFACTURING A corporation has three factories, each of which manufactures acoustic guitars and electric guitars The number of units of guitars produced at factory j in one day is represented by in the matri A 7 In Eercises 7, (a) write the system of linear equations as a matri equation, AX B, and (b) use Gauss-Jordan elimination on the augmented matri [A B] to solve for the matri X Find the production levels if production is increased by % MANUFACTURING A corporation has four factories, each of which manufactures sport utility vehicles and pickup trucks The number of units of vehicle i produced at factory j in one day is represented by in the matri a ij a ij A Find the production levels if production is increased by % 7 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,, and 7, respectively The numbers of bushels of peaches sent to the three outlets are, 7, and, respectively The profit per bushel for apples is $ and the profit per bushel for peaches is $ (a) Write a matri A that represents the number of bushels of each crop i that are shipped to each outlet j State what each entry of the matri represents (b) Write a matri B that represents the profit per bushel of each fruit State what each entry b ij of the matri represents (c) Find the product BAand state what each entry of the matri represents REVENUE An electronics manufacturer produces three models of LCD televisions, which are shipped to two warehouses The numbers of units of model i that are shipped to warehouse j are represented by in the matri A,,, The prices per unit are represented by the matri B $999 $999 Compute BA and interpret the result 9 INVENTORY A company sells five models of computers through three retail outlets The inventories are represented by S Model S A B C D E 9,,, 7 $999 Outlet The wholesale and retail prices are represented by Price a ij Wholesale Retail $ $ A $ $ B E T $ $ C Model $ $ D $ $ Compute ST and interpret the result a ij T

29 Section Operations with Matrices 97 7 VOTING PREFERENCES The matri P From is called a stochastic matri Each entry p ij i j represents the proportion of the voting population that changes from party i to party j, and p ii represents the proportion that remains loyal to the party from one election to the net Compute and interpret P 7 VOTING PREFERENCES Use a graphing utility to find P, P, P, P, P 7, and P for the matri given in Eercise 7 Can you detect a pattern as P is raised to higher powers? 7 LABOR/WAGE REQUIREMENTS A company that manufactures boats has the following labor-hour and wage requirements Labor per boat Department Cutting AssemblyPackaging Small Medium h h h S h h h h h h Large Wages per hour Plant A T $ $ $ B $ Cutting $ Assembly $ Packaging Compute ST and interpret the result 7 PROFIT At a local dairy mart, the numbers of gallons of skim milk, % milk, and whole milk sold over the weekend are represented by A A 7 R D I 7 Skim % Whole milk milk milk 9 R D To I 7 Department Friday Saturday Sunday Boat size 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 Selling price B $ $ $ Profit (a) Compute AB and interpret the result (b) Find the dairy mart s total profit from milk sales for the weekend 7 PROFIT At a convenience store, the numbers of gallons of 7-octane, 9-octane, and 9-octane gasoline sold over the weekend are represented by A A Octane The selling prices (in dollars per gallon) and the profits (in dollars per gallon) for the three grades of gasoline sold by the convenience store are represented by B Selling price B $ $ $ Profit (a) Compute AB and interpret the result (b) Find the convenience store s profit from gasoline sales for the weekend 7 EXERCISE The numbers of calories burned by individuals of different body weights performing different types of aerobic eercises for a -minute time period are shown in matri A Calories burned -lb person A 9 7 $ $ $ $ $9 $ Skim milk % milk Whole milk 7 9 Octane 9 -lb person Bicycling 9 Jogging 79 Walking Friday Saturday Sunday (a) A -pound person and a -pound person bicycled for minutes, jogged for minutes, and walked for minutes Organize the time they spent eercising in a matri B (b) Compute BA and interpret the result

30 9 Chapter Matrices and Determinants 7 HEALTH CARE The health care plans offered this year by a local manufacturing plant are as follows For individuals, the comprehensive plan costs $9, the HMO standard plan costs $, and the HMO Plus plan costs $9 For families, the comprehensive plan costs $7, the HMO standard plan costs $77, and the HMO Plus plan costs $ The plant epects the costs of the plans to change net year as follows For individuals, the costs for the comprehensive, HMO standard, and HMO Plus plans will be $9, $, and $997, respectively For families, the costs for the comprehensive, HMO standard, and HMO Plus plans will be $99, $7, and $7, respectively (a) Organize the information using two matrices A and B, where A represents the health care plan costs for this year and B represents the health care plan costs for net year State what each entry of each matri represents (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 net year (d) Suppose instead that the costs of the health care plans increase by % net year Write a matri that shows the new monthly payments EXPLORATION TRUE OR FALSE? In Eercises 77 and 7, determine whether the statement is true or false Justify your answer 77 Two matrices can be added only if they have the same order 7 Matri multiplication is commutative THINK ABOUT IT In Eercises 79, let matrices A, B, C, and D be of orders,,, and, respectively Determine whether the matrices are of proper order to perform the operation(s) If so, give the order of the answer 79 A C B C AB BC BC D CB D DA B BC DA 7 Consider matrices A, B, and C below Perform the indicated operations and compare the results A 7, B, C (a) Find A B and B A (b) Find A B, then add C to the resulting matri Find B C, then add A to the resulting matri (c) Find A and B, then add the two resulting matrices Find A B, then multiply the resulting matri by Use the following matrices to find AB, BA, ABC, and ABC What do your results tell you about matri multiplication, commutativity, and associativity? A 9 THINK ABOUT IT If a, b, and c are real numbers such that c and ac bc, then a b However, if A, B, and C are nonzero matrices such that AC BC, then A is not necessarily equal to B Illustrate this using the following matrices A 9 THINK ABOUT IT If a and b are real numbers such that ab, then a or b However, if A and B are matrices such that AB O, it is not necessarily true that A O or B O Illustrate this using the following matrices A 9 Let A and B be unequal diagonal matrices of the same order (A diagonal matri is a square matri in which each entry not on the main diagonal is zero) Determine the products AB for several pairs of such matrices Make a conjecture about a quick rule for such products 9 Let i and let A i,,, i B B B and,, B i C C i (a) Find A, A, and A Identify any similarities with i, i, and i (b) Find and identify B 9 Find two matrices A and B such that AB BA 9 CAPSTONE Let matrices A and B be of orders and, respectively Answer the following questions and eplain your reasoning (a) Is it possible that A B? (b) Is A B defined? (c) Is AB defined? If so, is it possible that AB BA?

31 Section The Inverse of a Square Matri 99 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 matrices Use inverse matrices to solve systems of linear equations Why you should learn it You can use inverse matrices to model and solve real-life problems For instance, in Eercise 7 on page 7, an inverse matri is used to find a quadratic model for the enrollment projections for public universities in the United States The Inverse of a Matri This section further develops the algebra of matrices To begin, consider the real number equation a b To solve this equation for, multiply each side of the equation by a (provided that a ) a b a a a b a b a b The number a is called the multiplicative inverse of a because a a The definition of the multiplicative inverse of a matri is similar Definition of the Inverse of a Square Matri Let A be an n n matri and let I n be the n n identity matri If there eists a matri such that A AA I n A A then A is called the inverse of A The symbol A is read A inverse Eample The Inverse of a Matri Alberto L Pomares/istockphotocom Show that B is the inverse of A, where A Solution and To show that B is the inverse of A, show that AB I BA, as follows AB BA As you can see, AB I BA This is an eample of a square matri that has an inverse Note that not all square matrices have inverses Now try Eercise B Recall that it is not always true that AB BA, even if both products are defined However, if A and B are both square matrices and AB I n, it can be shown that BA I n So, in Eample, you need only to check that AB I

32 Chapter Matrices and Determinants Finding Inverse Matrices If a matri A has an inverse, A is called invertible (or nonsingular); otherwise, A is called singular A nonsquare matri cannot have an inverse To see this, note that if A is of order m n and B is of order n m (where m n), the products AB and BA are of different orders and so cannot be equal to each other Not all square matrices have inverses (see the matri at the bottom of page ) If, however, a matri does have an inverse, that inverse is unique Eample shows how to use a system of equations to find the inverse of a matri Eample Finding the Inverse of a Matri Find the inverse of A Solution To find the inverse of A, try to solve the matri equation AX I for X A X I Equating corresponding entries, you obtain two systems of linear equations Linear system with two variables, Linear system with two variables, and and Solve the first system using elementary row operations to determine that and From the second system you can determine that and Therefore, the inverse of A is X A You can use matri multiplication to check this result Check AA A A Now try Eercise

33 Section The Inverse of a Square Matri TECHNOLOGY Most graphing utilities can find the inverse of a square matri To do so, you may have to use the inverse key Consult the user s guide for your graphing utility for specific keystrokes In Eample, note that the two systems of linear equations have the same coefficient matri A Rather than solve the two systems represented by and separately, you can solve them simultaneously by adjoining the identity matri to the coefficient matri to obtain A I This doubly augmented matri can be represented as A I By applying Gauss-Jordan elimination to this matri, you can solve both systems with a single elimination process R R R R So, from the doubly augmented matri A I, you obtain the matri I A A I I This procedure (or algorithm) works for any square matri that has an inverse A Finding an Inverse Matri Let A be a square matri of order n Write the n n matri that consists of the given matri A on the left and the n n identity matri I on the right to obtain A I If possible, row reduce A to I using elementary row operations on the entire matri A I The result will be the matri I A If this is not possible, A is not invertible Check your work by multiplying to see that AA I A A

34 Chapter Matrices and Determinants Eample Finding the Inverse of a Matri Find the inverse of A Solution Begin by adjoining the identity matri to A to form the matri A I Use elementary row operations to obtain the form I A, as follows R R R R R R R R R R R R So, the matri A is invertible and its inverse is A Confirm this result by multiplying A and A to obtain I, as follows I A WARNING / CAUTION Be sure to check your solution because it is easy to make algebraic errors when using elementary row operations Check AA Now try Eercise 9 The process shown in Eample applies to any n n matri A When using this algorithm, if the matri A does not reduce to the identity matri, then A does not have an inverse For instance, the following matri has no inverse A To confirm that matri A above has no inverse, adjoin the identity matri to A to form A I and perform elementary row operations on the matri After doing so, you will see that it is impossible to obtain the identity matri I on the left Therefore, A is not invertible I

35 Section The Inverse of a Square Matri The Inverse of a Matri Using Gauss-Jordan elimination to find the inverse of a matri works well (even as a computer technique) for matrices of order or greater For matrices, however, many people prefer to use a formula for the inverse rather than Gauss-Jordan elimination This simple formula, which works only for matrices, is eplained as follows If A is a matri given by A a c then A is invertible if and only if ad bc Moreover, if ad bc, the inverse is given by A b d ad bc d c b a Formula for inverse of matri A The denominator ad bc is called the determinant of the matri A You will study determinants in the net section Eample Finding the Inverse of a Matri If possible, find the inverse of each matri a b A B Solution a For the matri A, apply the formula for the inverse of a matri to obtain ad bc 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, A b For the matri B, you have ad bc which means that B is not invertible Now try Eercise Substitute for a, b, c, d, Multiply by the scalar and the determinant

36 Chapter Matrices and Determinants Systems of Linear Equations You know that a system of linear equations can have eactly one solution, infinitely many solutions, or no solution If the coefficient matri A of a square 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 System of Equations with a Unique Solution If A is an invertible matri, the system of linear equations represented by AX B has a unique solution given by X A B TECHNOLOGY To solve a system of equations with a graphing utility, enter the matrices A and B in the matri editor Then, using the inverse key, solve for X A B ENTER The screen will display the solution, matri X Eample Solving a System Using an Inverse Matri You are going to invest $, in AAA-rated bonds, AA-rated bonds, and B-rated bonds and want an annual return of $7 The average yields are % on AAA bonds, 7% on AA bonds, and 9% on B bonds You will invest twice as much in AAA bonds as in B bonds Your investment can be represented as y 7y z 9z z, 7 where, y, and z represent the amounts invested in AAA, AA, and B bonds, respectively Use an inverse matri to solve the system Solution Begin by writing the system in the matri form AX B y Then, use Gauss-Jordan elimination to find A A 7 Finally, multiply B by A on the left to obtain the solution X A B 7, 7 The solution of the system is, y, and z So, you will invest $ in AAA bonds, $ in AA bonds, and $ in B bonds Now try Eercise z,

37 Section The Inverse of a Square Matri EXERCISES See wwwcalcchatcom for worked-out solutions to odd-numbered eercises VOCABULARY: Fill in the blanks In a matri, the number of rows equals the number of columns If there eists an n n matri A such that AA I then A n A A, is called the of A If a matri A has an inverse, it is called invertible or ; if it does not have an inverse, it is called If A is an invertible matri, the system of linear equations represented by AX B has a unique solution given by X SKILLS AND APPLICATIONS In Eercises, show that B is the inverse of A 7 9 A A A A A A A A B In Eercises, find the inverse of the matri (if it eists),,, 7 B B B B,,, 7, B B 7, B In Eercises, use the matri capabilities of a graphing utility to find the inverse of the matri (if it eists) In Eercises, use the formula on page to find the inverse of the matri (if it eists)

38 Chapter Matrices and Determinants 9 In Eercises, use the inverse matri found in Eercise to solve the system of linear equations y y In Eercises 7 and, use the inverse matri found in Eercise to solve the system of linear equations 7 In Eercises 9 and, use a graphing utility to solve the system of linear equations using an inverse matri 9 7 In Eercises, use an inverse matri to solve (if possible) the system of linear equations y y y y y y In Eercises and, use the inverse matri found in Eercise 9 to solve the system of linear equations y z y z y z y z y z y z 9 9 y y y y y y y y y y 7 y z y z y z In Eercises 9, use the matri capabilities of a graphing utility to solve (if possible) the system of linear equations 9 y z y z 7y z In Eercises and, show that the matri is invertible and find its inverse sin cos A cos sin INVESTMENT PORTFOLIO In Eercises, consider a person who invests in AAA-rated bonds, A-rated bonds, and B-rated bonds The average yields are % on AAA bonds, 7% on A bonds, and 9% on B bonds The person invests twice as much in B bonds as in A bonds Let, y, and z represent the amounts invested in AAA, A, and B bonds, respectively y z 9 y z 7 y z 7y z y z 9y z 7 y 7y y Use the inverse of the coefficient matri of this system to find the amount invested in each type of bond Total Investment z total investment 9z annual return z Annual Return $, $7 $, $7 7 $, $ $, $, y z y z y z y z y 9z 7 9y 7z A sec tan y 7 y tan sec PRODUCTION In Eercises 9 7, a small home business creates muffins, bones, and cookies for dogs In addition to other ingredients, each muffin requires units of beef, units of chicken, and units of liver Each bone requires unit of beef, unit of chicken, and unit of liver Each cookie requires units of beef, unit of chicken, and units of liver Find the numbers of muffins, bones, and cookies that the company can create with the given amounts of ingredients

39 Section The Inverse of a Square Matri units of beef 7 units of beef units of chicken units of chicken units of liver units of liver 7 units of beef 7 units of beef 7 units of chicken 9 units of chicken 7 units of liver 9 units of liver 7 COFFEE A coffee manufacturer sells a -pound package that contains three flavors of coffee for $ French vanilla coffee costs $ per pound, hazelnut flavored coffee costs $ per pound, and Swiss chocolate flavored coffee costs $ per pound The package contains the same amount of hazelnut as Swiss chocolate Let f represent the number of pounds of French vanilla, h represent the number of pounds of hazelnut, and s represent the number of pounds of Swiss chocolate (a) Write a system of linear equations that represents the situation (b) Write a matri equation that corresponds to your system (c) Solve your system of linear equations using an inverse matri Find the number of pounds of each flavor of coffee in the -pound package 7 FLOWERS A florist is creating centerpieces for the tables at a wedding reception Roses cost $ each, lilies cost $ each, and irises cost $ each The customer has a budget of $ allocated for the centerpieces and wants each centerpiece to contain flowers, with twice as many roses as the number of irises and lilies combined (a) Write a system of linear equations that represents the situation (b) Write a matri equation that corresponds to your system (c) Solve your system of linear equations using an inverse matri Find the number of flowers of each type that the florist can use to create the centerpieces 7 ENROLLMENT The table shows the enrollment projections (in millions) for public universities in the United States for the years through (Source: US National Center for Education Statistics, Digest of Education Statistics) Year Enrollment projections 9 (a) The data can be modeled by the quadratic function y at bt c Create a system of linear equations for the data Let t represent the year, with t corresponding to (b) Use the matri capabilities of a graphing utility to find the inverse matri to solve the system from part (a) and find the least squares regression parabola y at bt c (c) Use the graphing utility to graph the parabola with the data (d) Do you believe the model is a reasonable predictor of future enrollments? Eplain EXPLORATION 7 CAPSTONE If A is a a b matri A c d, then A is invertible if and only if ad bc If ad bc, verify that the inverse is A TRUE OR FALSE? In Eercises 77 and 7, determine whether the statement is true or false Justify your answer 77 Multiplication of an invertible matri and its inverse is commutative 7 If you multiply two square matrices and obtain the identity matri, you can assume that the matrices are inverses of one another 79 WRITING Eplain how to determine whether the inverse of a matri eists If so, eplain how to find the inverse WRITING Eplain in your own words how to write a system of three linear equations in three variables as a matri equation, AX B, as well as how to solve the system using an inverse matri Consider matrices of the form A ad bc d c a a a b a (a) Write a matri and a matri in the form of A Find the inverse of each (b) Use the result of part (a) to make a conjecture about the inverses of matrices in the form of A PROJECT: VIEWING TELEVISION To work an etended application analyzing the average amounts of time spent viewing television in the United States, visit this tet s website at academiccengagecom (Data Source: The Nielsen Company) a nn

40 Chapter Matrices and Determinants THE DETERMINANT OF A SQUARE MATRIX What you should learn Find the determinants of 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, Eercises 9 on page show some types of determinants that are useful when changes in variables are made in calculus The Determinant of a Matri Every square matri can be associated with a real number called its determinant Determinants have many uses, and several will be discussed in this and the net section Historically, the use of determinants arose from special number patterns that occur when systems of linear equations are solved For instance, the system a b y c a b y c has a solution c b c b a b a b and provided that a b a b Note that the denominators of the two fractions are the same This denominator is called the determinant of the coefficient matri of the system Coefficient Matri A a a b b y a c a c a b a b Determinant deta a b a b The determinant of the matri A can also be denoted by vertical bars on both sides of the matri, as indicated in the following definition Definition of the Determinant of a The determinant of the matri A a a b b is given by deta A a a b b a b a b A In this tet, deta and are used interchangeably to represent the determinant of A Although vertical bars are also used to denote the absolute value of a real number, the contet will show which use is intended A convenient method for remembering the formula for the determinant of a matri is shown in the following diagram deta a a b b a b a b Matri Note that the determinant is the difference of the products of the two diagonals of the matri

41 Section The Determinant of a Square Matri 9 Eample The Determinant of a Matri Find the determinant of each matri a b c Solution a b c A B C deta detb detc 7 Now try Eercise 9 Notice in Eample that the determinant of a matri can be positive, zero, or negative The determinant of a matri of order is defined simply as the entry of the matri For instance, if A, then deta TECHNOLOGY Most graphing utilities can evaluate the determinant of a matri For instance, you can evaluate the determinant of A [ ] by entering the matri as [A] and then choosing the determinant feature The result should be 7, as in Eample (a) Try evaluating the determinants of other matrices Consult the user s guide for your graphing utility for specific keystrokes

42 Chapter Matrices and Determinants Minors and Cofactors To define the determinant of a square matri of order or higher, it is convenient to introduce the concepts of minors and cofactors Sign Pattern for Cofactors matri matri n n matri Minors and Cofactors of a Square Matri If A is a square matri, the minor M i j of the entry is the determinant of the matri obtained by deleting the ith row and jth column of A The cofactor C i j of the entry is In the sign pattern for cofactors at the left, notice that odd positions (where i j is odd) have negative signs and even positions (where i j is even) have positive signs Eample Finding the Minors and Cofactors of a Matri Find all the minors and cofactors of A Solution C i j ij M i j To find the minor M, delete the first row and first column of A and evaluate the determinant of the resulting matri M Similarly, to find M, delete the first row and second column M Continuing this pattern, you obtain the minors M M M Now, to find the cofactors, combine these minors with the checkerboard pattern of signs for a matri shown at the upper left C C C a i j,, M M M C C C Now try Eercise 9 M M M C C C a i j

43 Section The Determinant of a Square Matri The Determinant of a Square Matri The definition below is called inductive because it uses determinants of matrices of order n to define determinants of matrices of order n Determinant of a Square Matri If A is a square matri (of order or greater), the determinant of A is the sum of the entries in any row (or column) of A multiplied by their respective cofactors For instance, epanding along the first row yields A a C a C a n C n Applying this definition to find a determinant is called epanding by cofactors Try checking that for a matri A a a b b this definition of the determinant yields A a b a b, as previously defined Eample The Determinant of a Matri of Order Find the determinant of A Solution Note that this is the same matri that was in Eample There you found the cofactors of the entries in the first row to be C, C, and C So, by the definition of a determinant, you have A a C a C a C Now try Eercise 9 First-row epansion In Eample, the determinant was found by epanding by the cofactors in the first row You could have used any row or column For instance, you could have epanded along the second row to obtain A a C a C a C Second-row epansion

44 Chapter Matrices and Determinants When epanding by cofactors, you do not need to find cofactors of zero entries, because zero times its cofactor is zero a ij C ij C ij So, the row (or column) containing the most zeros is usually the best choice for epansion by cofactors This is demonstrated in the net eample Eample The Determinant of a Matri of Order Find the determinant of A Solution After inspecting this matri, you can see that three of the entries in the third column are zeros So, you can eliminate some of the work in the epansion by using the third column A C C C C Because C, C, and C have zero coefficients, you need only find the cofactor C To do this, delete the first row and third column of A and evaluate the determinant of the resulting matri C Delete st row and rd column Simplify Epanding by cofactors in the second row yields C 7 So, you obtain A C Now try Eercise 9 Try using a graphing utility to confirm the result of Eample

45 Section The Determinant of a Square Matri EXERCISES See wwwcalcchatcom for worked-out solutions to odd-numbered eercises VOCABULARY: Fill in the blanks Both deta and A represent the of the matri A The M ij of the entry a ij is the determinant of the matri obtained by deleting the ith row and jth column of the square matri The C of the entry of the square matri is given by ij ij Aa ij A M ij The method of finding the determinant of a matri of order or greater is called by SKILLS AND APPLICATIONS In Eercises, find the determinant of the matri In Eercises, use the matri capabilities of a graphing utility to find the determinant of the matri 9 7 In Eercises, find all (a) minors and (b) cofactors of the matri In Eercises, find the determinant of the matri by the method of epansion by cofactors Epand using the indicated row or column (a) Row (a) Row (b) Column (b) Column (a) Row (a) Row (b) Column (b) Column 7 7 (a) Row (a) Row (b) Column (b) Column 7 7 In Eercises 9, find the determinant of the matri Epand by cofactors on the row or column that appears to make the computations easiest 9 7

46 Chapter Matrices and Determinants In Eercises, use the matri capabilities of a graphing utility to evaluate the determinant In Eercises 7, find (a) (b) (c) AB, and (d) AB B, In Eercises 7 7, evaluate the determinant(s) to verify the equation 7 7 w c y cz 7 w y z 7 c w cw y z cy c w y z w y z y z w 7 7 A A A A, A A A A w cw a b a a y z,, a a b a y z B B,,,, A, B B B B B B, y z z y b a a b a b a In Eercises 77, solve for

47 Section The Determinant of a Square Matri In Eercises 9, evaluate the determinant in which the entries are functions Determinants of this type occur when changes of variables are made in calculus 7 e e e 9 9 ln ln ln y u v e e e e e EXPLORATION TRUE OR FALSE? In Eercises 9 and 9, determine whether the statement is true or false Justify your answer 9 If a square matri has an entire row of zeros, the determinant will always be zero 9 If two columns of a square matri are the same, the determinant of the matri will be zero 9 Find square matrices A and B to demonstrate that A B A B 9 Consider square matrices in which the entries are consecutive integers An eample of such a matri is 7 9 (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 9 WRITING Write a brief paragraph eplaining the difference between a square matri and its determinant 9 THINK ABOUT IT If A is a matri of order such that is it possible to find Eplain A, A? PROPERTIES OF DETERMINANTS In Eercises 97 99, a property of determinants is given ( A and B are square matrices) State how the property has been applied to the given determinants and use a graphing utility to verify the results 97 If B is obtained from A by interchanging two rows of or interchanging two columns of then A A, B A (a) 7 7 (b) 9 If B is obtained from A by adding a multiple of a row of A to another row of A or by adding a multiple of a column of to another column of then (a) A 7 A, B A (b) 99 If B 7 7 is obtained from A by multiplying a row by a nonzero constant c or by multiplying a column by a nonzero constant c, then (a) B c A 9 (b) 7 7 CAPSTONE If A is an n n matri, eplain how to find the following (a) The minor (b) The cofactor M ij C ij of the entry (c) The determinant of A of the entry a ij a ij In Eercises, evaluate the determinant CONJECTURE A triangular matri is a square matri with all zero entries either below or above its main diagonal A square matri is upper triangular if it has all zero entries below its main diagonal and is lower triangular if it has all zero entries above its main diagonal A matri that is both upper and lower triangular is called diagonal That is, a diagonal matri is a square matri in which all entries above and below the main diagonal are zero In Eercises, you evaluated the determinants of triangular matrices Make a conjecture based on your results Use the matri capabilities of a graphing utility to find the determinant of A What message appears on the screen? Why does the graphing utility display this message? A

48 Chapter Matrices and Determinants APPLICATIONS OF MATRICES AND DETERMINANTS MAFORD/istockphotocom 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 Eercise 9 on page 7, Cramer s Rule is used to find a quadratic model for the per capita consumption of bottled water in the United States 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 (7 7) 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 There, it was pointed out that the system a b y c a b y c has a solution c b c b a b a b and provided that a b a b Each numerator and denominator in this solution can be epressed as a determinant, as follows Coefficient Matri D a b a b a b a b For eample, given the system y y the coefficient matri, D, D, and are as follows Coefficient Matri y a c a c a b a b y a c a c a b a b a c a c b c b a a b a b c b c b a b a b Relative to the original system, the a denominator for and y is simply the determinant of the coefficient matri of the system This determinant is denoted by D The numerators for and y are denoted by D and D y, respectively They are formed by using the column of constants as replacements for the coefficients of and y, as follows D b D y c b c b D D a c a c D y D y c b

49 Section Applications of Matrices and Determinants 7 Cramer s Rule generalizes easily to systems of n equations in n variables The value of each variable is given as the quotient of two determinants The denominator is the determinant of the coefficient matri, and the numerator is the determinant of the matri 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 a a a b a a a b a a a b A A a a a a a a a a a a a a b b b a a a Cramer s Rule If a system of n linear equations in n variables has a coefficient matri A with a nonzero determinant A, the solution of the system is n A n A A, A, A, where the ith column of A i is the column of constants in the system of equations If the determinant of the coefficient matri is zero, the system has either no solution or infinitely many solutions A Eample Using Cramer s Rule for a System Use Cramer s Rule to solve the system of linear equations y y Solution To begin, find the determinant of the coefficient matri D Because this determinant is not zero, you can apply Cramer s Rule D D y Dy D So, the solution is and y Check this in the original system Now try Eercise 7

50 Chapter Matrices and Determinants Eample Using Cramer s Rule for a System Use Cramer s Rule to solve the system of linear equations y z z y z Solution To find the determinant of the coefficient matri epand along the second row, as follows D Because this determinant is not zero, you can apply Cramer s Rule D D y Dy D z D z D The solution is,, Check this in the original system as follows Check Now try Eercise??? Substitute into Equation Equation checks Substitute into Equation Equation checks Substitute into Equation Equation checks Remember that Cramer s Rule does not apply when the determinant of the coefficient matri is zero This would create division by zero, which is undefined

51 Section Applications of Matrices and Determinants 9 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 and is Area ± y, y,, y,, y y y where the symbol ± indicates that the appropriate sign should be chosen to yield a positive area Eample Finding the Area of a Triangle y (, ) (, ) (, ) FIGURE Find the area of a triangle whose vertices are,,,, and,, as shown in Figure Solution Let, y,, and Then, to find the area of the triangle, evaluate the determinant y, y,,, y, y y Using this value, you can conclude that the area of the triangle is Area Choose so that the area is positive square units Now try Eercise

52 Chapter Matrices and Determinants y (, ) (, ) (, ) FIGURE Lines in a Plane What if the three points in Eample 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 three collinear points,,,, and,, as shown in Figure The area of the triangle that has these three points as vertices is The result is generalized as follows Test for Collinear Points Three points, y and are collinear (lie on the same line) if and only if y,, y,, y y y Eample Testing for Collinear Points y 7 (7, ) (, ) 7 (, ) FIGURE Determine whether the points,,,, and 7, are collinear (See Figure ) Solution Letting, y,, and you have y, y,,, y 7,, y y 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 at these points is square units Now try Eercise 9

53 Section Applications of 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 Two-Point Form of the Equation of a Line An equation of the line passing through the distinct points and is given by, y, y y y y Eample Finding an Equation of a Line y (, ) (, ) FIGURE Find an equation of the line passing through the two points, and,, as shown in Figure Solution Let and Applying the determinant formula for the equation of a line produces, y,, y, y To evaluate this determinant, you can epand by cofactors along the first row to obtain the following y y So, an equation of the line is y Now try Eercise 7 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, and, is y y

54 Chapter Matrices and Determinants Cryptography A cryptogram is a message written according to a secret code (The Greek word kryptos means hidden ) Matri multiplication can be used to encode and decode messages To begin, you need to assign a number to each letter in the alphabet (with assigned to a blank space), as follows _ A B C D E F 7 G H 9 I J K L M N O P 7 Q R 9 S T U V W X Y Z Then the message is converted to numbers and partitioned into uncoded row matrices, each having n entries, as demonstrated in Eample Eample Forming Uncoded Row Matrices Write the uncoded row matrices of order 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 M E E T M E M O N D A Y Note that a blank space is used to fill out the last uncoded row matri Now try Eercise (a) To encode a message, use the techniques demonstrated in Section to choose an n n invertible matri such as A and multiply the uncoded row matrices by A (on the right) to obtain coded row matrices Here is an eample Uncoded Matri Encoding Matri A Coded Matri

55 Section Applications of Matrices and Determinants Eample 7 Encoding a Message Use the following invertible matri to encode the message MEET ME MONDAY A Solution The coded row matrices are obtained by multiplying each of the uncoded row matrices found in Eample by the matri A, as follows Uncoded Matri Encoding Matri A Coded Matri 77 So, the sequence of coded row matrices is 77 Finally, removing the matri notation produces the following cryptogram 77 Now try Eercise (b) For those who do not know the encoding matri A, decoding the cryptogram found in Eample 7 is difficult But for an authorized receiver who knows the encoding matri A, decoding is simple The receiver just needs to multiply the coded row matrices by A (on the right) to retrieve the uncoded row matrices Here is an eample Coded A Uncoded

56 Chapter Matrices and Determinants HISTORICAL NOTE Eample Decoding a Message Bettmann/Corbis Use the inverse of the matri 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 epressions for important military terms, such as iron-fish to mean submarine Without the Navajo Code Talkers, the Second World War might have had a very different outcome A to decode the cryptogram 77 Solution First find A by using the techniques demonstrated in Section A is the decoding matri Then partition the message into groups of three to form the coded row matrices Finally, multiply each coded row matri by A (on the right) Coded Matri Decoding Matri A Decoded Matri 77 So, the message is as follows M E E T M E M O N D A Y Now try Eercise CLASSROOM DISCUSSION 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