Home

x - The405

1. 2k tay F 2 2 5Sxt4y 2z2 4 20 3x 21y 297 1 2x 15y 2lz 21 x 2y 6c I 2x 5y 15z 4 3x y 327 6 a Il 6x 6y 122 l2ax 9 y z tn 2 x Dyer AX z 3 23 Think About lt Describe the row echelon form of an augmented matrix that corresponds to a system of linear equations that has a unique solution Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 650 8 Matrices and Determinants 24 Partial Fractions Write the partial fraction decom position for the rational expression IRI a CB E wiwt 4 1 wf 2 eae In Exercises 25 28 use the matrix capabilities of a graphing utility to reduce the augmented matrix and solve the system of equations 25 x 3y 2 26 x 3y 5 r y 2 4x y 2 ag Kor BY T T y z 4 Ay z 16 M w Ma Il i _ ta 2 td In Exercises 29 36 perform the matrix operations If it is not possible explain why 2 0 5 3 6 7 29 i i 3 A 2 7 l Ti s 4l Ri 2 6 0 4 By z 3 15 4
2. 9 Evaluate the determinant of the matrix 4 0 3 1 8 2 3 2 10 Use a determinant to find the area of the triangle in the figure Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it
3. on l 3 un un Ji 56 order of he answer A 2C 46 B 3C AB 48 BC BC D 50 CB D CAD 52 BCID D A 3B 54 BC DIA Factory Production A certain corporation has three factories each of which manufactures two products The number of units of product produced at factory J in one day is represented by a in the matrix 60 40 20 30 90 60 a Find the production levels if production is increased by 20 Hint Because an increase of 20 corre sponds to 100 20 multiply the given matrix by 1 2 Factory Production A certain corporation has four factories each of which manufactures two products The number of units of product produced at factory j in one day is represented by a in the matrix _ 100 90 70 30 40 20 60 ONY Find the production levels if production is increased by 10 Crop Production A fruit grower raises two crops which are shipped to three outlets The number of units of crop i that are shipped to outlet j is represent ed by a in the matrix LS a es me a 2s 150 100 The profit per unil ts represented by the matrix B 3 75 7 00 Find the product BA and state what each entry of the product represents FEyxploralion 8 2 Operations with Matrices 617 58 Revenue A manufacturer produces three models of a product which are shipped to two warehouses The number of units of model that are shipped to ware house is re
4. Note that the elementary row operation is written beside the row that is changed SRE EREE SERRE Bee 1SERR SORE ONTE BEBE OBB wa SERRE S822 SSPE Most graphing utilities can perform elementary row operations on matrices For instance on a 7 82 or T 83 you can perform the elementary row operation shown in Example 2 c as follows 1 Use the matrix edit feature to enter the matrix as A 2 Choose the row feature in the matrix math menu row 2 Al 1 3 The new row equivalent matrix will be displayed To do a sequence of row Operations use in place of A in each operation If you want to save this new matrix you must do this with separate steps Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 8 1 Matrices and Systems of Equations 593 In Example 2 of Section 7 5 you used Gaussian elimination with back substitution to solve a system of linear equations The next example demon strates the matrix version of Gaussian elimination The two methods are essentially the same The basic difference is
5. i 2 4 0 l Solution Note that this is the same matrix that was given in Example 2 There you found the cofactors of the entries in the first row to be Therefore by the definition of the determinant of a square matrix you have JA 4 Cy apli tgi First row expansion 0 1 2 5 1 4 l4 a 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 AoC 5 A570 55 A5 C 5 Second row expansion M2 1 4 28 14 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 632 amp Matrices and Determinants Study Tip Although most graphing utilities can calculate the determinant of a square matrix it is also important to know how to calculate them by hand When expanding by cofactors you do not need to find cofactors of zero entries because zero times its cofactor i5 zero a C O C 0 Thus the row or column containing the most zeros is usually the
6. 8 1 Matrices and Systems of Equations 599 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 For instance the following sequence of elementary row Operations on the matrix in Example 3 produces a slightly different row echelon form ly Be Ss 9 hs ot 2 fh f amp Ee V A p o og Mee L 17 S 5 8 17 Jw h i lo e3 0 nf l we be A Lal Ki 4 L e 3 le 5 tatg R R gt 0 O 2 4 L 3 O p 4 tet amp S aR 2 7 2 The corresponding system of linear equations 1s fs x SP yt az LA ha Try using back substitution on this system to see that you obtain the same solution that was obtained in Example 3 Group Activity Error Analysis One of your classmates has submitted the following steps for a solu tion of a system by Gauss Jordan elimination Find the error s in the solution and discuss how to explain the error s to your classmate Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content
7. A ee a as 0 7 1 5 2 5 P 261 el aT 2 LG i 0 In this chapter you studied several concepts that are required in the study of matrices and determinants and their applications You can use the following questions to check your understanding of several of these basic concepts The answers to these questions are given In Exercises 8 10 determine if the matrix operations a A 3B and b AB can be performed If not state why 8 10 11 12 14 15 eS 3 10 a 3 B 7 A aid E 2 A 7 Akb 2rd ifs es 115 30 SoA ag fides To A 7 2 s Paths 120 40 Under what conditions does a matrix have an inverse Explain the difference between a square matrix and its determinant Is it possible to find the determinant of a 4 x 5 matrix Explain What is meant by the cofactor of an entry of a matrix How is it used to find the determinant of the matrix Three people were asked to solve a system of equa tions using an augmented matrix Each person reduced the matrix to row echelon form The reduced matrices were Mea O ENS l 0 1 0 1 1 and Ja ha d 0 0 0 Could all three be right Explain ght 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s view has deemed that any suppressed content does not materially affect
8. B AB mxn nxp mxp t egual d order of AB EXAMPLE 7 Matrix Multiplication a a Lo he et Me a i w ale op oF ah Be og i l 2x3 3x3 2x3 N iN Hh O Tia 4 lo slom DI jae 4 2x2 ZX 2 2x32 i gisi Bp 4 hp od ow Rl fe 4 2x2 2x2 2x2 7 Le 3x1 Ixi 2 2 e i 2 3 1 2 3 I 2 3 3 xl ix3 3x3 sll hi f The product AB for the following matrices is not defined z 3 l 4 A 1 3 and B 0 D S 2 l 4 2 a 0 a2 3x4 Fy Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 612 ad nel 8 Matrices and Determinants ain We fd d M The general pattern for matrix multiplication is as follows To obtain the entry in the ith row and the jth column of the product AB use the ith row of A and the jth column of B bii biz a iA p i A ous na Dip b i by Ay eee E PaT by ba Das i m amp J bs bp Daa cae 2 fe Dip C j C x s 5 E Cc lj ip aj Zp Ci Cj ij Cin Cin Cm2 Cmj Cinp Properties of Matrix Multiplication Let A B and
9. B as follows Use a graphing utility to check this result 3 2 AB 4 2 E 5 0 1 3 4 Q G O 4 3 2 4 4 2 2 0 S 3 O 4 52 01 J y 4 6 15 10 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it My EXPLORATION Use a graphing utility to multiply the matrices l 2 A A and a f 3 Do you obtain the same result for the product AB as for the product BA What does this tell you about matrix multiplication and commu tativity Note In parts d and e of Example 7 note that the two products are different Matrix multiplication is not in general commutative That is for most matrices AB BA 8 2 Operations with Matrices 611 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 and the outside two indices give the order of the product as shown in the following diagram A
10. Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 642 8 Matrices and Determinants Cramer s Rule generalizes easily to systems of n equations in n variables The value of each variable 1s given as the quotient of two determinants The denom inator 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 x in the system AXi E aX a Dd aX F aX a b dX ayx d3 h is given by a a 28 5 Gy b ie _ Asl _ an ax b a E Eai JA du Ayn d laj d a ilag _ G3 ay ay If a system of n linear equations in n variables has a coefficient matrix A with a nonzero determinant A the solution is given by ay IA 1 tae OK ea er Al Al A where the th column of A is the column of constants in the system of equations If the coefficient matrix is zero the svstem has either no solu tion or infinitely many solutions xy EXAMPLE6 Using Cramer s Rule for a 3 x 3 System Use Cramer s Rule if possible to solv
11. __ee 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 x y and x gt y2 is given by X y l X 1 0 G Ws 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 verti cal line through 2 0 and 2 2 is X y l 0 l 0 a aa 2r 4 0 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 640 amp Matrices and Determinants EXAMPLE 4 amp Finding an Equation of a Line Figure 8 5 Find an equation of the line passing through the two points 2 4 and 1 3 as shown in Figure 8 5 solution Applying the determinant formula for the equation of a line produces X y l 2 4 H 0 E 3 l To evaluate this determinant you can expand by cofactors alon
12. a 0 0 7 3232 Sy l 0l gt G po l p At this point in the elimination process you can see that it is impossible to obtain the identity matrix on the left Therefore A is not invertible Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it AT EXPLORATION Use a graphing utility with matrix operations to find the inverse of the matrix Ges A gt 6l What message appears on the screen Why does the graphing utility display this message 8 3 The Inverse of a Square Matrix 623 The Inverse of a 2 2 Matrix Using Gauss Jordan elimination to find the inverse of a matrix works well even as 4 computer technique for matrices of order 3 x 3 or greater For 2 x 2 matrices however many people prefer to use a formula for the inverse rather than Gauss Jordan elimination This simple formula which works only for 2 x 2 matrices is explained as follows If A is a2 x 2 matrix given by pe A d then A is invertible if and only if ad be 0 If ad be 0 the inverse is given by l d p A l ad
13. ab a b a b provided that a b a b 0 Note that the denominator of each fraction is the same This denominator is called the determinant of the coefficient matrix of the system Coefficient Matrix Determinant a b A is d E T A The determinant of the matrix A can also be denoted by vertical bars on both sides of the matrix as indicated in the following definition Definition of the Determinant of a2 x 2 Matrix The determinant of the matrix os A La by is given by det A A a b 2 Note In this text det A and A 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 context will show which use is intended Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Note Notice in Example that the determinant of a matrix can be posi tive zero or negative 8 4 The Determinant of a Square Matrix 629 A convenient method for remembering the formula for the determinant of a 2 x 2 matrix is shown in the fol
14. angular as shown in the figure From the northern most vertex A of the region the distances to the other vertices are 25 miles south and 10 miles east for vertex B and 20 miles south and 28 miles east for vertex C Use a graphing utility to approximate the number of square miles in this region 20 Area ofa 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 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it In Exercises 31 26 use the determinant feature of a graphing utility to decide if the points are collinear 21 3 Se 0 Sap UE po an E j 6 1 10 2 23 ie 4 4 6 3 24 0 1 4 2 a 25 0 2 af is 1 1 6 26 2 3 3 3 5 1 2 In Exercises 27 32 use a determinant to find a
15. happen to be in reduced row echelon form The following matrices are not in row echelon form 2 e 0 2 0 f l 0 0 0 0 2 Every matrix has a row equivalent matrix that is in row echelon form For instance in Example 4 you can change the matrix in part e to row echelon form by multiplying its second row by 5 What elementary row operation could you perform on the matrix in part f so that it would be in row echelon form Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 8 1 Matrices and Systems of Equations 595 Gaussian Elimination with Back Substitution EXAMPLES Gaussian Elimination with Back Substitution Solve the system yt zg 2w 3 XK LZPp g g 2x 4y z2 3w 2 x 4y 7z w 19 Solution Study lip a Ky I 2 0 2 Gaussian elimination with R 10 l i z gt 3 n hack subetitivtte ts rafi TOR First column has leading s Bi pak ging WEDD RES IR 2 4 b gt 2 in upper Jett corner solving systems of linear equa a mo ay 9 tions by hand or with a c
16. jth column is denoted by the double subscript notation a A matrix having m rows and n columns is said to be of order m xn If m n the matrix is square of order n For a square matrix the entries ajj G55 433 are the main diagonal entries Examples of Matrices b Order x4 EXAMPLE 1 a Order x 2 S amp ce Order 2 x2 d Order 3 x2 5 d 2 0 0 os as Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 8 1 Matrices and Systems of Equations 591 A matrix derived from a system of linear equations each written in standard form with the constant term on the right is the augmented matrix of the system Moreover the matrix derived from the coefficients of the system but not including the constant terms is the coefficient matrix of the system System Augmented Matrix Coefficient Matrix e 4p t 3z 5 3 5 4 3 P 4y gS my ne Deed 3 3 i 2x 4z 6 2 D 6 2 0 4 Note Note the use of 0 for the missing v variable in the third equation and also note the fourth column of constant ter
17. learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 8 3 The Inverse of a Square Matrix 625 83 EXERCISES In Exercises 1 5 show that 8 is the inverse of A 1 2 33 ll 12 A gt es s s i l t d 13 14 1 i Zoey Ses Na Le 2 a 4 f x od 2 3 15 l 16 2 9 4 8 l 4 3 A i b 3 i 3 4 3 a gt 4 5 3 17 18 5 15 1 5 5 A g 2 0 l 4 A B 7 3 14 l l l 2 2 2 LAA EE te ia 35 go 2 3 7 9 5 A Q a R 3 3 6 5 j at 0 4 7 l 2 Q l 0 0 l 0 Q 2st Th I l 2 r H E 4 O 8 GO 6 amp 6 A 1 1l 7 BH 2 4 a W ag og Jo g 8 0 a Se 3 5 5 F 0 0 0 2 0 l 33 0 0 0 ee 3 0 0 l 0 0 4 0 a ay l z Ly 0 0 0 5 e Ti Ee 1 3 2 0 pie 0 2 4 6 p 4 9 5 6 ly Gs at 9 Sho De Sd ag Dawg E S Je 3 3 3 l 6 In Exercises 25 34 use the matrix capabilities of a W La 0 graphing utility to find the inverse of the matrix if it re od l 2 of exists 0 l E l 7 Cl 10 5 F Wee ae aa 25 po 1 26 5 B 3 0 i lt a 3 gt Z ie 0 l l 2 kat 2 rA 27 3 l 0 28 2 2 2 In Lwereises 9 24 lind the inverse of the matrix if it 2 0 A gt a 4 3 exists 0 1 0 2 03 2r 8 p o 2 29 Sh 0 2 02 30 0 3 0 9 F i 10 f 0 5 04 04 0 Q 5 Copyright 2010 Cengage Learning All Rights Reserved May not be copied s
18. of expansion by cofactors Expand using the indicated row or column 3 2 235 14 5 6 2 34 1 lt gf 2 6 6 8 4 7 8 a Row b Column 2 a Row 2 b Column 3 5 0 3 a Row 2 27 10 12 4 b Column 2 l 6 3 110 5 5 a Row 3 28 30 0 10 b Column 0 10 l 6 U 3 5 a Row 2 9 4 B3 6 b Column 2 o 7 4 8 6 0 2 10 8 s 7 a Row 3 30 4 0 a b Column q0 3 2 T l 0 3 2 In Exercises 31 40 find the determinant of the matrix Expand by cofactors on the row or column that appears to make the computations easiest lds 4 2 2 i 3 31 3 2 0 32 1 4 4 a 4 3 l 0 2 2 6 3 0 0 33 0 3 l 3 7 11 0 0 p 5 l 2 2 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 30 37 39 40 In Exercises 41 48 use the matrix capabilities of a EF ta a Wwe wm bo aa oc N __ a A N ocoo cow _NoOAW INNAD cococeoc NY oo orf WN 6 2 3 6 3 6 2 0 o ot Ml yg 0 7 0 3 0 6 pod 4 n a 3 lo o 2 2 3 2 Sei he A pia D ret ee s
19. subsequent rights restrictions require it 614 8 Matrices and Determinants Real Life EXAMPLE9 Softball Team Expenses Two softball teams submit equipment lists to their sponsors Women s Team Mens Team Bats 12 15 Balls 45 38 Gloves 15 17 Each bat costs 48 each ball costs 4 and each glove costs 42 Use matrices to find the total cost of equipment for each team Solution The equipment lists and the costs per item can be written in matrix form as I 15 E 45 38 and C 48 4 42 15 17 The total cost of equipment for each team is given by the product LZ 15 CE 48 4 42 45 38 1386 1586 ip 17 Thus the total cost of equipment for the women s team is 1386 and the total cost of equipment for the men s team is 1586 Group Activity Problem Posing Write a matrix multiplication application problem that uses the matrix ol ee ames iy 630 S01 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 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deem
20. that with matrices you do not need to keep writing the variables EXAMPLE 3 Using Elementary Row Operations Linear System Associated Augmented Matrix x 2y 3z 9 I2 3 9 Sy 4 A 0 o Ix Sy 52 17 Fee ee ee A Add the first equation to the Add the first row to the second equation second row R Ra x 2y 3z 9 I F B 9 y 3z 5 R tR a p 5 2x 5y 5z 17 E E A ge Ge Add 2 times the first equation Add 2 times the first row to to the third equation the third row 2R R4 Xx 2vy 3c 9 ee g y 3c 5 0 3 5 y f R Rh 10 al I a a Add the second equation to the Add the second row to the third equation third row R R3 x y 32 9 tf 2 3 9 y 3z 5 iret s 5 2z 4 R R 0 0 2 Multiply the third equation by 5 Multiply the third row by 5 x fy 3z 9 i 2 3 9 yr a9 4 l 3 i 5 z 2 Ro 0 8 3 2 At this point you can use back substitution to find that the solution is x 1 y 1 andz 2 as was done in Example 2 of Section 7 3 El Note Remember that you can check a solution by substituting the values of x vy and Z into each equation in the original system Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affe
21. use the graphing utility to find A x B and B x A Sre T A Xia 4xa 0 What can you conclude xi 3x5 0 y 3X55 From the first system you can determine that x 3 and xa 1 and from the second system you can determine that x 4 and x 1 Therefore the inverse of A is r A Tt i You can use matrix multiplication to check this result Check a SW al lo al 4 i ee e Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it My EXPLORATION Select two 2 x 2 matrices A and 5 that have inverses Enter them into your graphing utility and calculate AB Then calculate B A and A B Make a conjecture about the inverse of a product of two invertible matrices 8 3 The Inverse of a Square Matrix 621 Finding Inverse Matrices In Example 2 note that the two systems of linear equations have the same coefficient matrix A Rather than solve the two systems represented by jaa 3 yip tf s go ge gy eS 1 3 l separately you can solve them simultaneously by adjoining the iden
22. use the matrix capabilities of a graphing utility to write the matrix in reduced row echelon form ae be oe g 1 i 4 30 5 15 9 a 2 6 10 DP ft F g 1 2 A Be cs e g 4 8 I 14 3 1 8 BEE in a 2 10 In Exercises 33 36 write the system of linear equa tions represented by the augmented matrix Then use back substitution to find the solution Use variables x y and z Ty lt hie EB 3 i 34 3 k te aI 3 k TE S ee 4 _ 2 0 1 a3 resi Se f TA V3 9 T S a3 In Exercises 37 40 an augmented matrix that repre sents a system of linear equations in variables x y and z has been reduced using Gauss Jordan elimina tion Write the solution represented by the augmented matrix l 0 7 l f an Bi Pas f A w Og PoS 39 0 G P CE j 0 o0 3 40 0 j 0 0 In Exercises 41 56 solve the system of equations Use Gaussian elimination with back substitution or Gauss Jordan elimination 41 at 2y 7 42 2x 6y 16 xt y R 2x 3y 7 43 3x 5y 22 44 x 2y 0 3x 4y 4 x y 6 4x sy 32 3x 2y 8 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the ri
23. which the elementary row operations are applied EXAMPLE 8 4 A System with an Infinite Number of Solutions Solve the system 2x 4y 2z 0 3x t Dy Solution A ee a A RaM 2a p ga ea a g 3 Set oe ji 3R Ra lo 1 3 tow J boe hd i KR 0 O HR FeaT Vn 2 0 D lt 3 Shi The corresponding system of equations is x t3 2 y Be I Solving for x and y in terms of z you have x 5z 2 and y 3z Then letting z a the solution set has the form Sa 2 32 1a where is a real number Try substituting values for a to obtain a few solutions Then check each solution in the original system of equations E Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Note 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 given at the right to see that you obtain the same reduced row echelon form as in Example 7
24. 0 c Find the network fow pattern when x 1000 and x 600 500 600 S00 78 The flow of traffic in vehicles per hour through a network of streets is shown in the figure a Solve this system for the traffic flow represented bya i 1 2 5 4 5 b Find the traffic flow when x 200 and x 50 c Find the traffic flow when x 150 and x 0 150 200 79 The flow of traffic in vehicles per hour through a network of streets is shown in the figure a Solve this system for the traffic flow represented by xpi 1 2 3 4 b Find the traffic flow when x 0 100 li c Find the traffic flow when x Figure for 79 200 200 80 The fow of traffic in vehicles per hour through a network of streets is shown in the figure a Solve this system for the traffic flow represented Hys 2 3 4 5 b Find the traffic flow when x O and x 100 c Find the traffic flow when x x 100 600 300 ha 81 Chapter Opener Use the models on page 589 to estimate the men s and women s winning times in the 1000 meter speed skating events in the year 2002 82 Chapter Opener tf the models on page 589 contin ue to represent the winning times in the 1000 meter speed skating events in which winter Olympics will the women s time be less than the men s ume Review Solve Exercises 83 86 as a review of the skills and problem solving techniques you learned in pr
25. 28V 10V E 10V 65 Exploration Consider the matrices of the form ay 0 0 Wad Ai QO ay 0 on ee i A 0 0 ty Al sa ate 0 0 be pr HRY a Write a 2 x 2 matrix and a 3 x 3 matrix in the form of A Find the inverse of each b Use the result of part a to make a conjecture about the inverse of a matrix of the form of A Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 628 8 Matrices and Determinants 84 The Determinant of a Square Matrix The Determinant of a 2 2 Matrix Minors and Cofactors The Determinant of a Square Matrix I Triangular Matrices The Determinant of a 2 x 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 a x by c a x b3y cy has a solution given by Ciba Cb ajz AC a A y 2 Er ya a b
26. Bob Martin Allsport Matrices and Determinants OR 6 4 0 2 Women Men Time in seconds Year 0 lt gt 1980 Bonnie Blair won the 1000 meter women s speed skating event in the 1992 and 1994 winter Olympics These were the first times this event was ever won by an American a 589 righ ne third party content may be suppressed from the eBook and or eChapter s the right to remove additional content at any time if subsequent rights restrictions require it O Cle OnI rall learning experience Cengage Learning reserves 590 amp Matrices and Determinants Note The plural of matrix is matrices Note lt A matrix that has only one row is called a row matrix and a matrix that has only one column ts called a column matrix Matrices Elementary Row Operations J Gaussian Elimination with Back Substitution Gauss Jordan Elimination 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 matrix If m and n are positive integers an m x n read m by mn matrix is a rectangular array Gy Go Aiz 2s in Gy Gy aa Aan G3 G3 G33 ilan m rows iy 1 Ana Am3 Ginn k n columns in which each entry a of the matrix is a real number An m x n matrix has m rows horizontal lines and n columns vertical lines The entry in the ith row and
27. C be matrices and let c be a scalar 1 A BC AB C 2 A B C AB AC 3 A B C AC BC 4 c AB cA B A cB Associative Property of Matrix Multiplication Distributive Property Distributive Property Associative Property of Scalar Multiplication The n x n matrix that consists of 1 s on its main diagonal and 0 s elsewhere is called the identity matrix of order n and is denoted by oa i iu 10 FSD my ad i pp 0 0 Hentity matri Note that an identity matrix must be square When the order is understood to be n you can denote simply by Z IFA is an n x n matrix the identity matrix has the property that AZ A and I A A For example a 2 5 111 0 0 A110 l 2 3 0 0 and 0 3 Z Q ee 2 0 3 2 5 O 1 0 4 2 3 5 3 2 5 j l 1 QO 4 3 2 3 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Note The column matrix B is also called a constant matrix Its entries are the constant terms in the system of equations 8 2 J Operations with M
28. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 654 8 Matrices and Determinants B 11 CHAPTER TEST Take this test as you would take a test in class After you are done check your The Interactive CD ROM work against the answers given in the back of the book provides answers to the Chapter Tests and Cumulative Tests It also offers Chapter ys Pre Tests that test key skills In Exercises and 2 write the matrix in reduced row echelon form Use a graphing and concepts covered in pre utility to verify your result vious chapters and Chapter Post Tests both of which have I lt i 5 l W i 2 randomly generated exercises 1 16 gt 3 Miia l l t 3 with diagnostic capabilities s 3 _3 j f gt i 3 t 4 3 Use the matrix capabilities of a graphing utility to reduce the augmented matrix and solve the system of equations 4x 3y 2z2 14 p y 3n y 4dz 8 Figure for 4 4 Find the equation of the parabola y ax bx c that passes through the points in the figure Use a graphing utility to verify your result Find a A B b 3A and c 3A 2B zis 4 4 _f 4 1 6 asl 4 A s 0 A 6 Find AB if possible sa i 6 4 4 A 3 i1 Wits B8 J 3 2 2 QO 2 I oe 6 4 p 7 Find A forA 10 I 8 Use the result of Exercise 7 to solve the system 6x 4y 10 10x 5y 20 Figure for 10
29. RON What is the message 46 The following cryptogram was encoded with a2 x 2 matrix 3 25 Le 2 1S 15 3 14 8 13 38 19 19 19 37 16 The last word of the message is SUE What is the message Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it in the back of the book 1 Describe the three elementary row operations that can be performed on an augmented matrix 2 What is the relationship between the three elemen tary row operations on an augmented matrix and the operations that lead to equivalent systems of equa tions 3 In your own words describe the difference between a matrix in row echelon form and a matrix in reduced row echelon form In Exercises 4 7 the row echelon form of an augmented matrix that corresponds to a system of linear equations is given Use the matrix to determine whether the sys tem is consistent or inconsistent and if it is consistent determine the number of solutions i Vee aes 9 A WO ete 2 i aaute 80 0 it ete es 9 5 10 i 2 2 G2 O 8 i 2a Ss 9 6 10 1i 2 7 ie a 3
30. affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 620 8 Matrices and Determinants If a matrix A has an inverse A is called invertible or nonsingular otherwise A is called singular A nonsquare matrix cannot have an inverse To see this note that if A is of order m x n and B is of order n x m where m n the products AB and BA are of different orders and therefore cannot be equal to each other Not all square matrices possess inverses see the matrix at the bottom of page 622 If however a matrix does have an inverse that inverse is unique The following example shows how to use systems of equa tions to find the inverse of a matrix BBEE B ORTE r EXAMPLE2 4 Finding the Inverse of a Matrix HAE a8 an Find the inverse of Most graphing utilities have the hi g capability of finding the inverse A 4 of a square matrix For instance a ma to find the inverse of the matrix Solution Sa 5 To find the inverse of A try to solve the matrix equation AX for X A oy i s A K SAT ee 4 del i on a TI 82 or TI 83 enter the Ll 3 J L421 X22 0 matrix Then use the following xy 4x0 Xo 4 1 0 keystrokes 2 2 BOT ae ah aa 3X3 TX 19 FA 3X55 0 A ay ee eRe ye re Equating corresponding entries you obtain the following two systems of linear After You find ATE store it as B equations and
31. at any time if subsequent rights restrictions require it 600 8 J Matrices and Determinants Bl EXERCISES In Exercises 1 6 determine the order of the matrix 4 2 1 7 0 5 lt 3 amp FY 0 8 2 3 7 15 O 3 36 al De oy Bh Bi 3 by Ue T Eg ot In Exercises 7 10 form the augmented matrix tor the system of linear equations 7T 4x By S35 8 7x 4y 22 x 3y 12 5x 9y 15 9 x loys 2e 2 10 7x Seo 2 13 5x 3y4 4z 0 19x 8z 10 At YF in Exercises 11 14 write the System of linear iji lions represented by he augmented matrix Use ariables x and w l 2 7 fc Sey 0 Mt E 3 1 i f yo j 2 0 5 12 13 0 2 i T 6 3 0 2 9 12 3 0 i 0 4 2 18 5 2 10 l To o 0 L OF in Exercises 15 18 determine whether the matrix is in row echelon form If it is determine if it is also in reduced row echelon form De 18 Bie Gl Lb amp wo 2 B10 Fo eS Bole kh FN Go if e y tw p oD 2 4 0 Py 3 6 10 0 l 5 l 0 2 18 0 I e ig l 0 In Exercises 19 22 fill in the blanks using elementary row operations to form a row equivalent matrix E 93 yj TE 19 0 10 4 5 f 3 i tf 2 3 0 1 4 3 6 LI 1 l 2 8 wW go Eo E Rh Ba ea Sg SJ T 1 8 2u A ie O S l 0 5 1 1 3 0 3 2 4 1 tf 4 17 i a 4 GH 2 E 0 7 3 n a O b t en tila MD Pa 23 Perform the sequence of row operations on the matrix What did the operations accompli
32. atrices 613 Applications One application of matrix multiplication is representation of a system of linear equations Note how the system yy Xp T By Xy 3 B 3X 33 aX ba can be written as the matrix equation AX B where A is the coefficient matrix of the system and X and amp are column matrices Sa Ae Sa a bi ay n ly Xz by Ga Sad Og b A x xX B EXAMPLES 49 Solving a System of Linear Equations Solve the matrix equation AX B for X where Coefficient matrix Column matrix l d A 1 0 l 2 and B 4 2 4 2 Solution As a system of linear equations AX B is as follows Hp 4a Fe ES X 2y 4 2h 3k 2 a Using Gauss Jordan elimination on the augmented matrix of this system you obtain the following reduced row echelon matrix l 0 O0 E oq 0 l 0 2 o oO 1 3 Thus the solution of the system of linear equations is x 1 x 2 and x and the solution of the matrix equation is Ky X j 2 Xy l Use a graphing utility to verify that AX B Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if
33. be Be a Try verifying this inverse by multiplication Note The denominator ad bc is called the determinant of the 2 x 2 matrix A You will study determinants in the next section EXAMPLE 4 amp Finding the Inverse of a 2 x 2 Matrix If possible find the inverse of the matrix 3 e a pga 2s eee lee Solution aA a For the matrix A apply the formula for the inverse of a 2 x 2 matrix to obtain ad be 3 2 1 2 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 mh a multiplying by the scalar 4 as follows goat 2 2 f Ao 3l b For the matrix B you have Diw i ri bi ad be 3X2 1 6 0 which means that B is not invertible Sy Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 624 8 Matrices and Determinants The formula X A B is used on most graphing utilities to solve linear systems that have invertible coefficient matrices That is you e
34. best choice for expansion by cofactors This is demonstrated in the next example EXAMPLE 4 4 The Determinant of a Matrix of Order 4 x 4 Find the determinant of ij 3 0 l 0 2 A u 2 0 3 3 4 0 2 Solution After inspecting this matrix you can see that three of the entries in the third column are zeros Thus you can eliminate some of the work in the expansion by using the third column A 3 C O C OCC O C 4 Because C Cya 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 evalu ate the determinant of the resulting matrix il l 2 Ckh i r a BD 8 Delete ist row and 3rd column 3 4 2 l 2 0 2 3 Simplify 3 4 2 Expanding by cofactors in the second row yields the following C opl A 2 2 k a 5 4 2 2 vo gt ee 0 2 1 8 3 1 7 5 Thus you obtain A 3C 3 5 15 Note Try using a graphing utility to confirm the result of Example 4 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restricti
35. canned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 626 8 Matrices and Determinants 0 3 In Exercises 41 and 42 use an inverse matrix to solve 0 7 4 the system of linear equations Use the inverse matrix 31 0 3 found in Exercise 19 0 2 4 a ef yt 2 0 4A xt PE Sl 0 l 0 3x 5y 42 5 3x Sy 4z 3x y t z 2 3x 6y 5c 0 i 2 In Exercises 43 and 44 use an inyerse matrix and the matrix capabilities of a graphing utility to solve the system of linear equations Use the inverse matrix 1 2 3 5 2 3 33 1 a m a 3 F found in Exercise 33 Ss tp fees en 2a 0 4 Ss 4 FA om 0 2 1 7 2s Sey 2s Syl 3 GS F w x F 4n tn lin 2 35 IfA isa2 x 2 matrix given by H n ay a Ay _ 4 S I a Sey 7 A g d 2x Day By SKS 0 then A is invertible if and only if ad be 0 If ty Ag Pas Ly 3 ad be Q verify that the inverse is given by Fre i In Exercises 45 52 use an inverse matrix to solve if l possible the system of linear equations ad be e a 45 3x 4y 2 46 18 12y 13 36 Use the result of Exe
36. capabilities of a graphing utility to find f A aol aA a A27 O Ha A 4 ne 39 f x x S5x 2 A WARISE AS F j l 2 3 l 4 4i fis Ss Wx Sa 3 AHO 2 6 0 0D 5 42 f x x l0 2A A f 3 r 43 Think About It Ifa b and are real numbers such that c O and ac be 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 0 1 ha OR 2 F A b C oat ali of CP le al 44 Think About it fa and b are real numbers such that ab 0 then a 0 or b 0 However if A and B are matrices such that AB Q it is not necessarily true that A O or A O Illustrate this using the following matrices Beli Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Think About It ind B each of order 2 x 3 C of order 3 x 2 and D of order 2 x 2 Determine whether the matrices are of In Exercises 45 54 use matrices A proper order ty perform the operationis If so give its 45 47 49
37. ces and its applications are numerous This section and the next two intro duce some fundamentals of matrix theory It is standard mathematical conven tion to represent matrices in any of the following three ways 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 brackets such as aj b or ey 3 A matrix can be denoted by a rectangular array of numbers such as Gi Ai Qiz Ay gt A gt da3 PA An A a a3 A395 33 a Mn am d m2 d m3 73 Ginn Two matrices A a and B b are equal if they have the same order m x n anda b for lt i lt m and 1 lt j lt n In other words two matrices are equal if their corresponding entries are equal EXAMPLE1 Equality of Matrices Solve for d 9 amp and a in the following matrix equation ki se 2 H a n 3 0 Solution Because two matrices are equal only if their corresponding entries are equal you can conclude that Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions
38. ct the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 594 Some graphing utilities such as the 7 85 TI 92 and HP 48G can automatically transform a matrix to row echelon form and reduced row echelon form Read your user s manual to see if your calculator has this capability If SO use it to verify the results in this section 8 Matrices and Determinants The last matrix in Example 3 is said to be in row echelon form The term echelon refers to the stair step pattern formed by the nonzero elements of the matrix To be in this form a matrix must have the following properties A matrix in row echelon form has the following properties 1 All rows consisting entirely of zeros occur at the bottom of the matrix 2 For each row that does not consist entirely of zeros the first nonzero entry is called a leading 1 3 For two successive nonzero rows the leading 1 in the higher row is farther to the left than the leading in the lower row A matrix 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 1 EXAMPLE4 Row Echelon Form The following matrices are in row echelon form ae a it Eg 0 0 lt 5 y ith 6p i g 0 0 b o 1 Oo ww bo The matrices in b and d also
39. determinant of a 3 x 3 triangular matrix is the product of its main diagonal entries yee A 0 dy claz Aid anda Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 634 8 I Matrices and Determinants B4 EXERCISES In Exercises 1 16 find the determinant of the matrix i a k p oo E D lt 3 3 1 Sa lig a gy 3 2 2 as l TARA 7 6 o 4 3 A a 2 6 af lt 3 his al oy Pees 2 1 O 2 2 3 Mid es a 12 1 1 O e S44 oO 1 4l 60 3 7 fai 28 13 0 0 o0 wW 3 o O 4 6 3 E 43 1 2 5 hid v i iS 0 3 4 16 4 1 0 i amp 3 a air 5 In Exercises 17 20 use the matrix capabilities of a graphing utility to find the determinant of the matrix 03 02 03 Si Bal Wise 3 17 02 02 63 18 0 3 02 0 2 04 04 03 0 5 04 04 E E ey ae 4 19 3 6 6 M10 S 2 a H 8 0 0 2 In Exercises 21 24 find all a minors and b cofac tors of the matrix Pan Be 4 11 0 af 4 af 9 3 g 8 e 9 4 a 3 2 H 24 7 6 0 i 3 6 6 T 6 In Exercises 25 30 find the determinant of the matrix by the method
40. dinate measures the horizontal distance from the player in feet and the y coordinate is the height of the ball in feet a Find the equation of the parabola y ax bx c that passes through the three points b Use a graphing utility to graph the parabola Approximate the maximum height of the ball and the point at which the ball strikes the ground c Find analytically the maximum height of the ball and the point at which it strikes the ground Height in feet 20 40 60 B0 100 Horizontal distance in feet Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 604 8 Matrices and Determinants Network Analysis In Exercises 77 80 answer the questions about the specified network In a network it is assumed that the total fow into each junction is equal to the total Naw out of the junction 77 Water is flowing through a network of pipes in thou sands of cubic meters per hour See figure a Solve this system for the water Now represented by x 8 1 2 3 4 5 6 7 b Find the network flow pattern when x x
41. ditive identity for the set of all mm x n matrices For example the following matrices are the additive identities for the set of all 2 x 3 and 2 x 2 matrices fo Jeo 0 Q 0 S Zero 2 x 3 matrix Zero 2 x 2 matrix The algebra of real numbers and the algebra of matrices have many similarities For example compare the following solutions Real Numbers m x n Matrices Solve for x Solve for X xta b X A B85 x at a b a X A t A B A x 0 bhb a X O 8B A x b a X B A EXAMPLE5 Solving a Matrix Equation Solve for X in the equation 3X A B where aol E a EN Et 4 A y 3 and aa a i Solution Begin by solving the equation for X to obtain 3X B A gt X B A Now using the matrices A and B you have x e fi a jd mg H E L a 4 6 2 3 W A j Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 610 8 Matrices and Determinants Note The definition of matrix multi plication indicates a row by column multiplication where the entry in the ith row and jt
42. e a 0 A 8 ES S u S GIE we 32 2 p e illa 0 j 3 6 4 33 3 5 2 0 J Q y 34 s e a0 2 pl 53 gost Sie 36 ET ie ui E bh PA p sd T In Exercises 37 40 use a graphing utility to perform the matrix operations a ST _ gf oe F l av at 3 3 s 7 5 3 ri 4 2 38 5 7 2 4 6 11 a 2 3 4 3 5 6 39 i1 7 f s a a as AD g Ea s 0 40 aati l 5 2 arer A G S In Exercises 41 44 solve for X given 4 E A 1 5 and B 2 1i an 3 4 4 41 X 3A 2B 43 3X 2A 8 42 6X 4A 3B 44 2A 5B 3X 45 Write the system of linear equations represented by the matrix equation Ss Be 3 1 itty 2f 46 Write the matrix equation AX B for the following system of linear equations a 3y z 10 2x 3y 32 2 4x 2y 32 hr pa In Exercises 47 50 use a graphing utility to find the inverse of the matrix if it exists aa A G 3 10 af _ as 3 7 ify ig be Mair 1S 49 1 0 2 x 2 I 1 18 16 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remo
43. e measured in amperes Solve the system of equations In Exercises 69 74 find the specified equation that passes through the points Use a graphing utility to verify your result 69 Parabola y ax brte 99 3 20 71 Cubic y axt h exrtd ij T f 2 73 Quartic y axt dr e i l 3 5 1 3 AV AP 2 0 5 0 0 70 Parabola on yHax bhx e 10 1 9 72 Cubic y ax bh r d 5 1 0 875 74 Quartic y actt dxrt e lt 2 10 a9 T1 0 5 2 6 3 2 5 75 Reading a Graph 76 8 1 Matrices and Systems of Equations 603 The bar graph gives the value y in millions of dollars for new orders of civil jet transport aircraft built by U S companies in the years 1990 through 1992 Source Aerospace Industries Association of America a Find the equation of the parabola that passes through the points Let f 0 represent 1990 b Use a graphing utility to graph the parabola c Use the equation in part a to estimate y in 1993 y Value in millions of dollars Ra Fi a2 Year 0 lt 1990 Mathematical Modeling After the path of a ball thrown by a baseball player is videotaped it is ana lyzed on a television set with a grid covering the screen The tape is paused three times and the position of the ball is measured each time The coordinates are approximately 0 5 0 15 9 6 and 30 12 4 The x coor
44. e the following system of linear equations z 4 A fr S e y 3z Solution Using the matrix capabilities of a graphing utility to evaluate the determinant of the coefficient matrix A you find that Cramer s Rule cannot be applied because A 0 iy Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 8 5 Applications of Matrices and Determinants 643 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 O assigned to a blank space as follows y _ 9 I8 R I A lQ J I9 5 2 B 1l K 20 T s C 2 L 21 U 4 D 13 M 22 V S E I4 N 23 W 6 F I5 0O 24 X T G l6 P 25 Y H I7 Q 26 Z Then the message is converted to numbers and partitioned into uncoded row matrices cach having n entries as demonstrated in Example 7 EXAMPLE7 Forming Uncoded Row Matrices Write the uncoded row matrices of orde
45. east squares regression line y a bt tor this data is found by solving the system l3a 91b 1107 Ola 819b 8404 7 where t 1 represents 1981 Source National Association of Realtors a Use a graphing utility to solve this system b Use a graphing utility to graph the regression line c Interpret the meaning of the slope of the regres sion line in the context of the problem d Use the regression line to estimate the median price of homes in 1995 ras te Year 1 1981 ais Saves onama a A S hes g 3 A In Exercises 81 84 use a determinant to find the area of the triangle with the given vertices 81 1 0 5 0 5 8 82 4 0 4 0 0 6 g4 3 1 4 3 4 2 in Exercises 85 88 use a determinant to find an equa tion of the line through the given points 85 4 0 4 4 86 2 5 6 1 87 3 3 4 1 88 0 8 0 2 0 7 3 2 89 Verily that aii ai diy az 432 23 dy FC p TF Co la T C3 a p a Oy Stia cig Gy n 93 T an dyl Alay Sap ag CG Ra iy 90 Circuit Analysis Consider the circuit in the figure The currents and in amperes are given by the solution of the system of linear equations Use the inverse of the coefficient matrix of this system to tind the unknown currents i AE l l Q Al F 10L je 101 21 6 l2 6v 91 Think About It If A is a3 x 3 matrix and A 2 what is the val
46. ect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 8 5 Applications of Matrices and Determinants 641 th b ay cb cb Ce Ba Aja C y 5 p T ier m a b db ad 2 Gites Gash a By ls by il b Relative to the original system the denominator for x and y is simply the deter minant of the coefficient matrix of the system This determinant is denoted by D The numerators for x and y are denoted by D and D respectively They are formed by using the column of constants as replacements for the coefficients of x and y as follows Coefficient Matrix D iy D Us b Gls b Cs b as Cs EXAMPLE5 Using Cramer s Rule for a 2 x 2 System Use Cramer s Rule to solve the following system of linear equations 4x 2y 10 3x 3y 11 Solution To begin find the determinant of the coefficient matrix 4 2 3J i 5 720 6 14 Because this determinant is not zero you can apply Cramer s Rule to find the solution as follows We P e M L SO ASh SB R j 14 T Ai H D B3 nj 4 30_ 4 p SB th Ty Therefore the solution is x 2 and y 1 Check this in the original system Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part
47. ed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 8 2 Operations with Matrices 607 The symbol A represents the scalar product 1 A Moreover if A and B are of the same order A B represents the sum of A and 1 B That is A 6B A US Subtraction of matrices EXAMPLE 3 Scalar Multiplication and Matrix Subtraction For the following matrices find a 3A b B and c 3A B 2 2 4 2 0 A 3 0 li and B I 4 3 2 l 2 3 2 Solution 2 2 4 a 3A 3 3 ab Scalar multiplication 2 l 2 3 2 32 3 4 a 3 4D 3 1 Multiply each entry by 3 A EXPLORATION Select two 3 x 2 matrices A and B Enter them into your graphing zS utility and calculate A B and B A What do you observe 9 w 3 Simplify m on a 2 0 0 Oa ee rn ne What do you observe 3 p 0 4 3 Multiply each entry by 1 3 2 6 G 12 2 0 0 c 34 B 9 T E y l 4 3 Matrix subtraction 6 3 6 z 3 2 4 6 12 10 4 6 Subtract corresponding entries 7 0 4 E It is often convenient to rewrite the scalar multiple cA by factoring c out of l every entry in the matrix For instance in the following example the scalar 5 has been factored o
48. ed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Be EXERCISES 8 2 Operations with Matrices 615 In Exercises 1 4 find x and y 16 4 16 4 2x 1 4 ZE i 3 13 15 3x 0 0 2 3ye p ete 3 Ix 6 8 i 4 Il 2y 2x i 8 T A yer 2 7 2 Wilk In Exercises 5 10 find a A and d 3A 2B op B b A B ce 3A 5 A _ Tie g _ 3 2 6 A d d 6 li i ia A 2 4 Esje 3 3 5 10 eee T2 lt 3 4 sas i B l Ej sgu e Wh of sasl bie 10 g 1 1 i y a So fd of SG S 3 4 1 A 2 B 5 2 In Exercises 11 14 solve for X given 2 1 0 A l 0 and B 2 3 4 4 ll X 3A 2B 13 2X 3A B 12 14 2X 2A B 2A 4B 2X In Exercises 15 20 find a AB b BA and if possi ble c A Note A AA 16 A 17 21 A 22 A 23 ies II oS tJ ofi 2 f4 2 a Eea 4 3 I AN 3 ie SH l 1 2 j 3 f3 J l 0 Al Bo l4 0 6 gs 0 2 2 B 3 7 3 it 3 5 B 5 lo 7 Q 3 ol B 0 2 0 0 0l B 0 7 0 aa 2 g o o 0 B 3 i ag s i l 3 B _ 7 S B 2 1 2 i B 3 0 2 2 0 9 7 rail 6 0 0 5 0 0 Copyright 2010 Cengage Learning All Right
49. ents tn Exercises 57 60 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 twice as much in B bonds as in A bonds Let x y and z represent the amounts invested in AAA A and B bonds respectively x y z total investment 0 065x 0 07y 0 092 annual return 2y z Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond 57 Total investment 25 000 Annual return 1900 58 Total investment 45 000 Annual return 3750 59 Total investment 12 000 Annual return 835 60 Total investment 500 000 Annual return 38 000 8 3 The Inverse of a Square Matrix 627 61 Essay Write a brief paragraph explaining the advantage of using an inverse matrix to solve the systems of linear equations in Exercises 37 44 62 True or False Multiplication of an invertible matrix and its inverse is commutative Give an exam ple to demonstrate your answer Circuit Analysis In Exercises 63 and 64 consider the circuit in the figure The currents and J in amperes are given by the solution of the system of lin ear equations 21 41 E I 41 E where E and E are voliages Use the inverse of the coefficient matrix of this system to find the unknown currents for the given voltages Ii 63 E 64 E l4 V E
50. evi ous sections Graph the function and check each graph with a graphing utility 83 f x F 84 elx 85 h x logy 1 86 f a v 2 l Did 34 Inx Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it D E 8 2 Operations with Matrices 605 Operations with Matrices A British mathematician Arthur Cayley invented matrices around 1858 Cayley was a Cambridge University graduate and a lawyer by profession His ground break ing 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 Equality of Matrices Matrix Addition and Scalar Multiplication Matrix Multiplication Applications Equality of Matrices In Section 8 1 you used matrices to solve systems of linear equations Matrices however can do much more than this There is a rich mathematical theory of matri
51. ew has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 8 5 Applications of Matrices and Determinants 645 EXAMPLE9 4 Decoding a Message 2 2 Use the inverse of the matrix A 1l l 3 to decode the cryptogram i 4 I3 260 21 3S 34 1 16 25 42 5 m20 56 24 23 Th Solution Partition the message into groups of three to form the coded row matrices Then multiply each coded row matrix by A on the right Coded Matrix Decoding Matrix A Decoded Matrix 1 10 8 13 26 21 1 6 5 13 5 5 0 i 1 1 10 8 33 53 12 1 6 5 20 0 13 0 L 1 10 8 18 23 42 1 6 5 5 0 13 0 1 i 1 10 8 5 20 56 1 6 5 15 4 4 a Aail 10 8 24 23 77 1 S 1 25 Q 0 1 i Thus the message is as follows 13 5 5 20 0 13 5 0 13 15 14 4 1 25 0 MEET M E ow wy ay Group Activity Cryptography Create your own numeric code for the alphabet such as on page 643 and use it to convert a message of your own into numbers Create an invertible n x n matrix A to encode your message Exchange your numeric code encoded message and matrix A with another group Find the necessary decoding matrix and decode the message you received Copyright 2010 Cengage Learning All Rights Reserved May not be copied sca
52. g the first row to obtain the following l i y 1 ai i Tus i 4 x 3y 10 0 Theretore an equation of the line ts x 3y 10 0 Note There are a variety of ways to check that the equation of the line in Example 4 is correct You can check it algebraically using the techniques you learned in Section P 3 or you can check it graphically by plotting the points and graphing the line in the same viewing rectangle 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 You will now 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 ax biy c Hla A T bsy Cs has a solution given by ae ciba cb aC a5 and y provided that a b a b 0 Each numerator and denominator in this solu tion can be expressed as a determinant as follows Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially aff
53. ght to remove additional content at any time if subsequent rights restrictions require it 602 8 Matrices and Determinants 45 8x 4y Sx 2y 47 x 2y 2x 4y 49 x i e a y 2x 2y l 24 ot 24 Bi 14 7x Sy 6 Sl gey 5 x 2 gt aa lie 52 2x 3 4x 3y 5 8x 9y 15 9 Sac XE Ze 8 3x Ty 6z 26 54 4x 12y 7s 20w WE Qy Sz 2Bw 55 2 P 56 x 2y 0 2x 4y 0 In Exercises 57 62 use the matrix capabilities of a graphing utility to reduce the augmented matrix and solve the System of equations Ste Ak Sy IZ x y Az 2x Sy 202 SE BY ae Bz 58 2 10y x Sy x Sy 3x Sy as Pa J gt WN WwW I tjd Sa co 59 60 61 63 64 65 67 ax y t lw 6 3x 4y w X S 22 Ow 3 lt r Zoe w 3 x 2y 2z2 4w Il 3x 6y 5z 12w 30 x pee 0 62 x 2y z 3w 0 2x 3y z 0 i w 3X Sy 2 0 y zt2w 0 Think About lt The augmented matrix represents a system Of linear equations in variables x y and z that has been reduced using Gauss Jordan elimina tion Write a system of equations with nonzero coefficients that is represented by the reduced matrix The answer is not unique l 0 3 i 2 4 0 oO 0 0 Think About It a Describe the row echelon form of an augmented matrix that corresponds fo a system of linear equations tha
54. h column of the product AB is obtained by multiplying the entries in the ith row of A by the cor responding entries in the jth column of B and then adding the results Example 6 illustrates this process HEEE SESS E Tilli BBB e Leae EE oon j Some graphing utilities such as the 7 82 and TI 83 are able to add subtract and multiply matri ces If you have such a graphing utility enter the matrices A F and ee l a 2 B 4 2 0 l 2 3 and use the following keystrokes to find the product of the matrices a x B You should get 8 4 a lt 25 16 5 Matrix Multiplication The third basic matrix operation is matrix multiplication At first glance the following definition may seem unusual You will see later however that this definition of the product of two matrices has many practical applications If A a is an m x n matrix and B b is an n x p matrix the product AB is an m x p matrix AB c b iH n where cj apb apby agb gt a Finding the Product of Two Matrices EXAMPLE 6 Find the product AB where 3 3 gt A 4 2 and B B 5 0 3 Solution First note that the product AB is defined because the number of columns of A is equal to the number of rows of B Moreover the product AB has order 3 x 2 and 1s of the form 3 3 ay SP o Sia S 0 Ot lisa ge To find the entries of the product multiply each row of A by each column of
55. if both products are defined However if A and B are both square matrices and AB it can be shown that BA I Hence in Example 1 you need only to check that AB A The Inverse of a Matrix Finding Inverse Matrices The Inverse of a2 x 2 Matrix Systems of Linear Equations The Inverse of a Matrix This section further develops the algebra of matrices To begin consider the real number equation ax b To solve this equation tor x multiply both sides of the equation by a provided that a 0 ax b alax a b 1 x a b x a b Il The number a is called the multiplicative inverse of a because a a The definition of the multiplicative inverse of a matrix is similar Let A be an n x n matrix If there exists a matrix A such that AA I A A A is called the inverse of A EXAMPLE 1 The Inverse of a Matrix Show that is the inverse of A where i 2 iL 2 B s A E j and f aig Solution To show that B is the inverse of A show that AB J BA as follows ft am Tita 2 2 PQ AB E ii iq Bee i 3 fs 1 ary Was j k gt 4 nan 2I P a HPs 2 2 70 0 i li 4 Aren 23 0 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially
56. ist wants to arrange a dozen 76 77 78 flowers consisting of two varieties carnations and roses Carnaticns cost 0 75 each and roses cost 1 50 each How many of each should the florist use so that the arrangement will cost 12 00 Mixture Problem One hundred liters of a 60 acid solution is obtained by mixing a 75 solution with a 50 solution How many liters of each must be used to obtain the desired mixture Fitting a Parabola to Three Points Find an equa tion of the parabola y ax bx that passes through the points 1 2 0 3 and 1 6 Break Even Point A sm ll business invests 25 000 in equipment to produce a product Each unit of the product costs 3 75 to produce and is sold for 5 25 How many items must be sold before the business breaks even Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 652 8 Matrices and Determinants 79 Data Analysis The median prices y in thousands of dollars of one family houses sold in the United States in the years 1981 through 1993 are shown in the figure The l
57. ition of More Than Two Matrices By adding corresponding entries you obtain the following sum of four matrices 1 1 fo 2 2 2 1 1 3 1 y 2 4 2 l a E LLI EEE HN AEE E E BEBE 0 E n Honn SES on _ Aw EE ID RE L CE i memini Most graphing utilities can add and subtract matrices and multiply matrices by scalars For instance on a 77 82 or TI 83 you can find the sum of 2 3 4 ael d and B 5 a by entering the matrices and then using the following keystrokes A 8 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Note The algebra of real numbers und the algebra of matrices also have important differences Which will be discussed later 8 2 Operations with Matrices 609 One important property of addition of real numbers is that the number 0 is the additive identity That is 0 e for any real number c For matrices a similar property holds That is if A is an m x n matrix and O is the m x n zero matrix consisting entirely of zeros then A O A In other words O is the ad
58. lowing diagram s P Note that the determinant is given by the difference of the products of the two diagonals of the matrix EXAMPLE 1 The Determinant of a 2 x 2 Matrix Find the determinant of each matrix ee a 3 gael b B 5 ele 2 Tee 4 2o Solution 2 3 a devia 20 i 3 44 3 7 2 I b det B 209 41 4 4 0 R 3 c det C y F 0 4 2 0 3 3 The determinant of a matrix of order x 1 is defined simply as the entry of the matrix For instance if A 2 det A Most graphing utilities can evaluate the determinant of a matrix For instance on a 77 82 or 77 83 you can evaluate the determinant of 2 3 ath by entering the matrix as A and then choosing the det feature in the matrix math menu det A The result should be 7 as in Example l a Try evaluating determinants of other matrices What happens when you try to evaluate the determinant of a nonsquare matrix Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require i
59. mbine these minors with the checkerboard pattern of signs shown at the left for a 3 x 3 matrix to obtain the following Cy 1 S3 2 Cy 4 Ci 72 Ga 4 Ca 8 C3 7 Ce 3 Caa ll i Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 8 4 f The Determinant of a Square Matrix 631 The Determinant of a Square Matrix The following definition is called inductive because it uses determinants of matrices of order n 1 to define the determinant of a matrix of order n If A is a square matrix of order 2 x 2 or greater the determinant of A is the sum of the entries in any row or column of A multiplied by A tiia 4412 their respective cofactors For instance expanding along the first row yields Note Try checking that for a 2 x 2 matrix this definition yields as previously defined JA ay Cyy Paba a C Ln Applying this definition to find a determinant is called expanding by cofactors EXAMPLE 3 The Determinant of a Matrix of Order 3 x 3 Find the determinant of 0 2 l AS Jg
60. move additional content at any time if subsequent rights restrictions require it 622 8 Matrices and Determinants Verify the computations in Example 3 with your graphing utility Enter the 3 x 6 matrix A J and row reduce ittothe matrix Z gt A as follows bn SO e GY 1 oO 1 O 6 2 3 0 oO 4 i 0 M N l 0 l 0 0 Lee Se What happens if you try this method to find the inverse of RS A 3 1 2 3 EXAMPLE 3 Finding the Inverse of a Matrix b gt i 0 Find the inverse of A 1 i p 2 3 Solution Begin by adjoining the identity matrix to A to form the matrix f e He u fA AlS fi 0 1 0 l 0l m S e p 2 wo Using elementary row operations to obtain the form Z7 A results in l 0 0 3 l Wop he g ee es OL 0 0 2 Qh l Therefore the matrix A is invertible and its inverse is eas Sa l A 3 3 ER Try using a graphing utility to confirm this result by multiplying A by A to obtain 7 E The process shown in Example 3 applies to any n x n matrix A If A has an inverse this process will find it IfA does not have an inverse the process will tell us so For instance the following matrix has no inverse l 2 0 A 3 l 2 Io og To confirm that matrix A above has no inverse begin by adjoining the identity matrix to A to form l 2 0 0 0 Aeros e ae GB iy ihe At 2 e amp Df 1 Then use elementary row operations to obtain mie
61. ms in the augmented matrix When forming either the coefficient matrix or the augmented matrix of a system you should begin by vertically aligning the variables in the equations and using O s for the missing variables Given System Line Up Variables Form Augmented Metrix E sy gt 8 xh By 9 3 0 9 y 42 2 y AZ 2 0 j x 52 0 x 35 0 H 3 o Elementary Row Operations In Section 7 3 you studied three operations that can be used on a system of linear equations to produce an equivalent system I Interchange two equations 2 Multiply an equation by a nonzero constant 3 Add a multiple of an equation to another equation In matrix terminology these three operations correspond to elementary row operations An elementary row operation on an augmented matrix of a given system of linear equations produces a new augmented matrix 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 l Interchange two rows 2 Multiply a row by a nonzero constant 3 Add a multiple of a row to another row Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppre
62. n equa tion of the line through the points 27 0 0 5 3 28 0 0 2 2 29 t 4 ah 2 1 30 10 7 2 7 31 3 3 3 1 32 5 4 6 12 In Exercises 33 and 34 find x such that the points are collinear 33 2 3 4 i 5 34 6 2 3 x 2 2 In Exercises 35 and 36 find the uncoded 1 x 3 row matrices for the message Then encode the message using the matrix Message Matrix j 0 35 TROUBLE IN RIVER CITY i 2 3 2 l 36 PLEASE SEND MONEY A a 3 2 In Exercises 37 40 write a cryptogram for the mes sage using the matrix LSO A 3 TFT OL 4 7 37 LANDING SUCCESSFUL 38 BEAM ME UP SCOTTY 8 5 Applications of Matrices and Determinants 647 39 HAPPY BIRTHDAY 40 OPERATION OVERLORD In Exercises 41 and 42 use A to decode the eryp fogram i 3 A 4 11 21 64 112 25 50 29 53 23 46 40 75 55 92 i 0 42 A 0 f 2 3j 9 9 38 19 19 28 8 19 80 25 41 64 21 31 9 5 4 In Exercises 43 and 44 decode the cryptogram by using the inverse of the matrix 1 2 2 A 3 7 9 1 4 7 a3 20 17 15 12 36 25 5 0 143 181 Ah 15 9 39 01 112 106 mla 73 M 1 24 29 65 144 172 45 The following cryptogram was encoded with a 2 x 2 matrix 8 21 13 16 13 13 5 10 3 25 3 19 1 6 20 40 18 18 1 16 The last word of the message is
63. nned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 646 8 Matrices and Determinants 11 EXERCISES In Exercises 1 10 use a determinant to find the area of the triangle with the giyen vertices l 2 3 E 3 0 0 1 5 3 1 5 0 3 i oh 4 a 7 4 5 6 1 7 9 8 oh pee 4 x 5 9 3 5h 2 Bh mi 10 2 4 1 5 3 2 In Exercises H and 12 find a value of x such that the triangle has an area of 4 I 3 1 0 2 F 12 4 2 3 5 1 x In Exercises 13 16 use Cramer s Rule to solve if pos sible the system of equations 13 3x 4y 2 14 0 4x 0 8y Sx 3y 4 0 2x 0 3y 2 2 Bok pe gah 16 4 2 oe 2 2x 2y Sz 10 2x 2y 5z 16 5x 2y 6z 1 ax 3y 2z 4 In Exercises 17 and 18 use a graphing utility and Cramer s Rule to solve if possible the system of equations Li ga ce Satan 5 3x 5y 9z 5x 9y 17z 4 Il l 18 23x 3y S5z 4 3x 35y 9z 7 5x 9y 17z I bJ ua 19 Area of a Region A large region of forest has been infected with gypsy moths The region is roughly tri
64. nned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Figure 8 4 You can use the following steps on a 77 82 or TI 83 graphing cal culator to check whether three points are collinear 1 Plot the points by entering Pt On x y for each point Pt On can be found in the DRAW POINTS menu 2 Draw a line from the two farthest points by entering Line Xis Yita Yah Use the steps above to check the results of Example 3 Explain how the graph shows that the points are not collinear Why is Step 2 important in determining if points are collinear 8 5 Applications of Matrices and Determinants 639 EXAMPLE 3 Testing for Collinear Points Determine whether the points 2 2 1 1 and 7 5 lie on the same line See Figure 8 4 Solution Letting x 2 2 Y2 1 1 and x3 y3 7 5 you have ty Fe J l Mm Ya a L Lg l l l l l l l 2 1 2 1 1 1 2 4 2 6 1 2 6 Because the value of this determinant is net zero you can conclude that the three points do not lie on the same line a c T aaa e a
65. nter the n x n coefficient matrix A and the n x 1 column matrix B The solution X is given by AT B Note Use Gauss Jordan elimination or a graphing utility to verify A for the system of equations in Example 5 Group Activity You know that a system of linear equations can have exactly one solution infinitely many solutions or no solution If the coefficient matrix A of a square system a system that has the same number of equations as variables is invert ible the system has a unique solution which is given as follows A System of Equations with a Unique Solution If A is an invertible matrix the system of linear equations represented by AX B has a unique solution given by X A B EXAMPLE5 Solving a System of Equations Using an Inverse Use an inverse matrix to solve the system 2o 39 eS 3x 3yt z2 1 2x 4y 2z2 2 l 0 2 X A B 1 0 1 6 2 3 2 Thus the solution is x 2 y 1 and z 2 i Finding an Inverse Matrix Use a graphing utility to decide which of the following matrices is are invertible NSAR A lie A b B 2 9 5 d s ae ae ee Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall
66. om the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 636 8 J Matrices and Determinants In Exercises 63 66 find a A bi B ec AM und id AB 0 Zi I 3 n ae 4 n 4 zj 1 2 aj 2 8 4 i 2 0 0 65 A BU ES is S i Do 0 0 3 z Q l te 4 66 A 1 2 B 0 l 3 3 3 2 67 Exploration Find square matrices A and B to demonstrate that A B A B 68 Exploration Consider square matrices in which the entries are consecutive integers An example of such ad matrix is 4 5 7 BQ ie ab 12 a Use a graphing utility to evaluate four determi nants of this type Make a conjecture based on the results b Verify your conjecture 69 Essay Write a brief paragraph explaining the differ ence between a square matrix and its determinant Think About It If A is a matrix of order 3 x 3 such that A 5 is it possible to find 2A Explain 70 ln iexercises 71 74 a property of determinants is viven State how the property has heen applied to the given determinants and use a praphing utility lo veri I v the resulis 71 If A and amp are square matrices and is obtained from A by interchanging two rows of A or interchanging t
67. omputer For this algorithm the order in 2 0 2 at the pri E row opera I D lons are performed 1s Important z l BA COUR Tie CST We suggest operating from left to 2R Ry 10 0 3 3 6 below its leading right by columns using elemen Ri Faam 0 a 2 tary row operations to obtain be et Clk Ae i 2 f 0 a zeros in all entries directly below a lanek ri 3 the leading I s Second column has eros 0 0 x 3 6 below its leading E 2 0 2 i l 2 3 Third column has z ros 3K Ei 0 2 below its leading 1 0 0 3 39 A 0 2 ef tw 3 3 Fourth column has d U s 2 leading oie N 8 OF 3 The matrix is now in row echelon form and the corresponding system is tay k 2 y z2 2w 3 z w 2 w 3 Using back substituuion you can determine that the solution ts x 1 y 2 z 1 andw 3 Check this in the original system of equations Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 596 8 f Matrices and Determinants Gaus
68. ons require it My EXPLORATION The formula for the determinant of a triangular matrix discussed at the right is only one of many properties of matrices You can use a computer or calculator to discover other properties For instance how is cA related to A How are A and B related to AB Group Activity 8 4 The Determinant of a Square Matrix 633 Triangular Matrices Evaluating determinants of matrices of order 4 or higher can be tedious There is however an Important exception the determinant of a triangular matrix A square matrix is upper triangular if it has all zero entries below its main diagonal and lower triangular if it has all zero entries above its main diagonal A matrix that is both upper and lower triangular is called diagonal That is a diagonal matrix is one in which all entries above and below the main diagonal are zero Upper Triangular Matrix Lower Triangular Matrix aly lis ty ei ly il 0 0 0 0 i 13 i a3 TET VET ilsa 0 arm A 0 ily tH be Ci SET 45 iqq T o o 0 0 es Ga ct th Eha ik tf To find the determinant of a triangular matrix of any order simply form the product of the entries on the main diagonal EXAMPLE 5 4 The Determinant of a Triangular Matrix if of G A 0 0 a 9 iis 12 a 5 5 0 2 2 0 3 l 5 3 z 0 0 0 0 0 0 or AB b g 0 O OF G 2 4 2 48 0 0 0 4 0 Write an argument that explains why the
69. presented by a in the matrix 5 000 4 000 6 000 10 000 8 000 5 000 The price per unil is represented by the matrix B 20 50 26 50 29 50 Compute BA and interpret the result In Esercises 59 amd 60 let i 59 Consider the matrix an i a i Find A A and A Identify any similarities with i i and i Find and identify A for the matrix Inventory Levels A company sells five models of computers through three retail outlets The inven tories are given by Model A 8B oD E 3 2 23 w a S 10 2 3 4 3 2 Outliei 2 L 3 23 The wholesale and retail prices are given by T Price Wholesale Retail 840 1100 1200 1350 B 1450 1650 Model 2650 3000 D 3050 3200 E Compute ST and interpret the result Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 8 3 8 3 The Inverse of a Square Matrix 619 The Inverse of a Square Matrix Note The symbol A is read A inverse Note Recall that it is not always true that AB BA even
70. r 1 x 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 13 5 5 20 0 13 5 0 13 1S 14 4 1 25 0 MEE T M E M ONDAY Note that blank space is used to fill out the last uncoded row matrix 2 To encode a message choose an n x n invertible matrix A and multiply the uncoded row matrices by A on the right to obtain coded row matrices Here is an example Uncoded Matrix Encodine Matrix A Coded Matrix i 3 13 5 oS j i 1 OB I 2 21 p 4 This technique is further illustrated in Example 8 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 644 8 Matrices and Determinants EXAMPLE 8 4 Encoding a Message Use the following matrix to encode the message MEET ME MONDAY y lt 2 ss r C 3 I i 4 Solution BEEE EREE 8 The coded row matrices are obtained by multiplying each of the uncoded row H i x EBEE Ss ERA 7 matrices found in Example 7 by the matrix A as follow
71. rcise 35 to find the inverse of ae each matrix ser ay 4 30x 24y 23 5 aT 47 0 4x 0 8y 1 6 48 13x 6y 17 a a a 3 Ix 4y 5 26x l2y 8 7 0S 49 3x 6y 6 50 3x 2y 1 b LR a h 5 6x l4y 11 2x 10v 6 Sl 4r yt 2 35 In Exercises 37 40 use an inverse matrix to solve the 2x 2y 3z 10 system of linear equations Use the inverse matrix found in Exercise 11 Mm Zhoz l 37 x y 5 38 x 2y 0 on 44 2y 32 2 at ap 10 2x SY 3 2x 2y 5z 16 39 x 2Zy 4 4 x 2y Bx Sy z A a Sy 2 i SY 2 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it In Exercises 53 56 use the matrix capabilities of a graphing utility to solve if possible the system of lin ear equations o3 Sk Sy 2z 2 54 2x 3y 5z2 4 Bet 2 Sz SS 3x boy Gime 7 x Ty 8 4 5x 9y 17z 13 SD x Sy 2w 41 Sek y w 3 4x z 2w J2 P w 56 2x 5y w il x 4y 22 Ze 7 2x 2y 5z w 3 x 3w 1 Bond Investm
72. require it 606 8 Matrices and Determinants Matrix Addition and Scalar Multiplication You can add two matrices of the same order by adding their corresponding entries If A a and B b are matrices of order m x n their sum is the ij m x n matrix given by A B a bj The sum of two matrices of different orders is undefined SHES ERRE H EXAMPLE 2 amp Addition of Matrices DENNE Beare EAE HE BE gE ql a 0 j ot ea g g gp pj 1 F he P BL Lr sk s 1 1 2 3 0 1 1 2 T Most graphing utilities can per form matrix addition and scalar multiplication If you have such a b graphing utility duplicate the matrix operations in Examples 2 i 0 and 3 Try adding two matrices of a different orders such as GEH ET AESA 2 QO 3 T f and d The sum of pe C4 S 0 l 6 A 4 0 i and B 1_ 3 What error message does your a aa 2 2 4 ake okt h utility display AA pea In work with matrices numbers are usually referred to as scalars In this text scalars will always be real numbers You can multiply a matrix A by a scalar c by multiplying each entry in A by c If A a is an m x n matrix and c is a scalar the scalar multiple of A by c is the m x n matrix given by cA cay Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppress
73. s HEHA RE EEEE Uncoded Matrix Encoding Matrix Coded Matrix An efficient method for encoding L 2 2 the message at the right with your is a 5 1 3 13 26 21 graphing utility is to enter A as a i s j i 3 x 3 matrix Let B be the 5 x 3 a e matrix whose rows are the uncod 2 2 ed row matrices 20 O 13 1 1 3 B3 53 12 13 5 5 i lt 20 Q 13 2 2 B a ERY Bani 5 O0 13 1 3 18 23 42 15 14 4 I 4 The product BA gives the coded n5 4 4 3 5 20 56 row matrices it Sb iow i 9 2 iG 25 QO 1 I 3 24 23 77 p 4 Thus 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 lp 26 21 23 38 12 18 23 42 S205 24 23 TF For those who do not know the matrix A decoding the cryptogram found in Example 8 is difficult But for an authorized receiver who knows the matrix A decoding is simple The receiver need only multiply the coded row matrices by A on the right to retrieve the uncoded row matrices Here is an example 03 26 2 a 13 5 4 a a E ca T Coded Uncoded Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial revi
74. s Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 616 8 J Matrices and Determinants 10 26 A 15 B 6 2 6 0 5 16 11 4 27 10 0 3 B s 16 4 0 0 4 0 0 3 ft 2 i g 28 A 13 Sad s 5 In Exercises 29 34 use the matrix capabilities of a graphing utility to find AB 5 6 3 lL 2 29 A 2 5 Ik B 8 l 4 i 2 5 4 2 9 II I Fia 13 a id Wel B fs fe A 8 I5 16 30 A 3 8 6 8 a 12 If 2 Gi 4 8 loa 15 14 S Ne J0 8 4 10 3 age TR 3 A 21 s 6l 3 F gl 15 32 14 0 5 1 6 _f 9 10 38 18 he Neh 50 250 _ 52 85 27 45 40 35 60 82 Il po EA I5 18 cT AEE 12 8 22 F 33 ngl B 8 16 A In Exercises 35 38 find matrices A X and B such that the system of linear equations can be written as the matrix equation AX B Solve the system of equations Use a graphing utility to check your result 35 x y 4 3 x x 2y 3z 9 2x y 0 I APS 2S 8 Bt Oy amp eae 37 oer Sy 5 38 xt y 3z l x 4y 10 u By ro z 0 In Exercises 39 42 use the matrix
75. sh Law s 2 4 3 f a Add 2 times Row to Row 2 b Add 3 times Row to Row 3 c Add 1 times Row 2 to Row 3 d Multiply Row 2 by lt e Add 2 times Row 2 to Row 1 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it The Interactive CD ROM contains step by step solutions to all 8 1 J Matrices and Systems of Equations 601 odd numbered Section and Review Exercises It also provides Tutorial Exercises which link to Guided Examples for additional help 24 Perform the sequence of row operations on the matrix What did the operations accomplish 7 2 4 L a Add Row 3 to Row 4 b Interchange Rows 1 and 4 c Add 3 times Row 1 to Row 3 d Add 7 times Row to Row 4 e Multiply Row 2 by 5 f Add the appropriate multiples of Row 2 to Rows 1 3 and 4 In Exercises 25 28 write the matrix in row echelon form Remember that the row echelon form of a matrix is not unique 0 5 25 2 I 2 10 3 6 Ff ia 2 3 3 FF 2 i 3 8 2a 10 In Exercises 29 32
76. sian Elimination with Back Substitution I 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 is inconsistent EXAMPLE 6 A System with No Solution Solve the system a Ge a a x z 2x 3y 32 4 3x 2y z 1 Solution Pe h 2 4 pit ooo g ia ii 6 Ri R gt 0 i 1 i 2 2 4 4 R R 0 i 4 E 4 es I 3R R 0 5 7 ll rh i l i oe fT t F kg oo ge 0 of F 2 ti R R gt 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 ts inconsistent You can see why this is true by converting back to a system of linear equations x y 2z 4 p g5 Ff 0 2 a gt Because the third equation is not possible the system has no solution Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights
77. some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it For a demonstration of a graphi cal approach to Gauss Jordan elimination on a 2 x 3 matrix see the graphing calculator program for this section in the appendix Note Which technique do you prefer Gaussian elimination or Gauss Jordan elimination 8 1 Matrices and Systems of Equations 597 Gauss Jordan Elimination m l With Gaussian elimination elementary row operations are applied to a matrix to obtain a row equivalent row echelon form A second method of elimina tion called Gauss Jordan elimination after Carl Friedrich Gauss 1777 1855 and Wilhelm Jordan 1842 1899 continues the reduction process until a reduced row echelon form is obtained This procedure is demon strated in the following example EXAMPLE 7 amp Gauss Jordan Elimination Use Gauss Jordan elimination to solve the system x tj 9 y 4 Sy 4 2x Sy 52 17 Solution In Example 3 Gaussian elimination was used to obtain the row echelon form fi 2 3 9 Goi 5 rt TARE 2 Now rather than using back substitution apply additional elementary row operations until you obtain a matrix in reduced row echelon form To do this you m
78. ssed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 592 amp Matrices and Determinants The Interactive CD ROM shows every example with its solution clicking on the Try t button brings up similar problems Guided Examples and Integrated Examples show step by step solutions to additional examples Integrated Examples are related to several concepts in the section The Interactive CD ROM offers graphing utility emulators of the 7 82 and T 83 which can be used with the Examples Explorations Technology notes and Exercises Although elementary row operations are simple to perform they involve a lot of arithmetic Because it is easy to make a mistake we suggest that you get in the habit of noting the elementary row operations performed in each step so that you can go back and check your work EXAMPLE 2 Elementary Row Operations a Interchange the first and second rows Original Matrix New Row Equivalent Matrix 0 3 4 R 2 0 3 2 0 3 Ri 0 3 4 a eg 4 l Ma eae 4 1 b Multiply the first row by 5 Original Matrix New Row Egquivalent Matrix 2 4 6 2 Rael f 2 a1 l r 3 0 E Mr 0 Jo l 2 a 2 l 2 c Add 2 times the first row to the third row Original Matrix New Row Equivalent Matrix 2 4 3 l 2 4 3 0 7 Ss 0 q 2 2 l ay aR tR 0 TE og
79. t 630 8 I Matrices and Determinants Minors and Cofactors To define the determinant of a square matrix of order 3 x 3 or higher it is convenient to introduce the concepts of minors and cofactors T eee ee ee ee J Saj T iL Minors and Cotactors of a square Me ar a Tg te es Ck A ee a i LT If A is a square matrix the minor M of the entry a is the determinant of the matrix obtained by deleting the ith row and jth column of A The cofactor C of the entry a is given by ya W Sign Pattern for Cofactors w A SS EXAMPLE 2 amp Finding the Minors and Cofactors of a Matrix Find all the minors and cofactors of 3 6 3 matrix 0 2 l Pf tat a A 3 r 2 DHI Pa M 4 0 1 T sp Solution 4 4 matrix To find the minor M delete the first row and first column of A and evaluate the determinant of the resulting matrix _ 1 ep M 0 lt 1 002 1 Ss eb ie Ge E OR fF Fee Similarly to find M delete the first row and second column 3 3 Si eee R nxn matrix 4 fae Mi i i 3 1 42 5 4 l Continuing this pattern you obtain the following minors a P kr A Note In the sign pattern for cofactors My E Mi Miz 4 above notice that edd positions where My 2 My 4 Mpy itj is odd have negative signs and Mi 5 M 2 Mn 6 even positions Where i j is even have positive signs Now to find the cofactors co
80. t 9 0 0 2 4 3 2 2 6 3 a 0 0 2 graphing utility to evaluate the determinant 41 j0 43 45 47 48 3 8 O N N A A Ww ooo OWY o od 4 l 6 0 14 5 4 2 12 l 8 6 Q 0 2 2 8 4 0 2 0 T ose 2 3 0 0 3 0 0 0 0 0 0 5 8 4 9 7 ee 3 0 44 2 5 2 4 0 3 4 a AE re 0 7 0 e 4 1 0 x 2 0 0 ay 2 WEF 0 0 0 0 MEE 0 4 0 4 l oc m GO 1 oc _ OO hv 8 4 The Determinant of a Square Matrix 635 In Exercises 49 52 evaluate the determinants lo veri fy the equation 49 ci y z w x en i exl _ x y ce y Z s1 w x i x cw y Zz s Zr oy Y cw cx In Exercises 53 and 54 evaluate the determinant to verify the equation l E i 53 1 y yt y x z xz y l AE 2 a b a a 54 a a b a 6 amp 3a tbd a a a b In Exercises 55 and 56 solve for x y 2 3 x 2 eee Shs 0 55 wa 56 In Exercises 57 62 evaluate the determinant where the entries are functions Determinants of this type occur in calculus 4u 3x 3y ajea api e e e xe oP 2e 3e o0 e 1 xe x Inx x xinx 61 L itt 62 l l inx Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed fr
81. t 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 Borrowing Money A small corporation borrowed 1 500 000 to expand its product line Some of the money was borrowed at 8 some at 9 and some at 12 How much was borrowed at each rate if the annual interest was 133 000 and the amount bor rowed at 8 was 4 times the amount borrowed at 12 Borrowing Money A small corporation borrowed 500 000 to expand its product line Some of the money was borrowed at 9 some at 10 and some at 12 How much was borrowed at each rate if the annual interest was 52 000 and the amount borrowed at 10 was 2 times the amount borrowed at 9 Partial Fractions Write the partial fraction decom position for 407 x 1 e 1 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 68 Electrical Network The currents in an electrical network are given by the solution of the system i h 0 21 24 F 2 4 8 where and ar
82. terminants EXAMPLE 2 Finding the Area of a Triangle ee oe wt eit Find the area of the triangle whose vertices are 3 7 64 4 and 8 5s Figure 8 2 as Shown in Figure 8 2 7 Solution Let xp y 3 xs yo 64 45 and x y 8 5 Then to find the area of the triangle evaluate the determinant i l A 2 xy yy 3 7 x Ys 1 6 45 I i Xs Va SS Ds Using the matrix capabilities of a graphing utility you find the value of the determinant to be 65 76 Now you can use this value to conclude that the area of the triangle 1s Area 5 65 76 32 883 Figure 8 3 Lines in the Plane Suppose the three points in Example 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 0 1 2 2 and 4 3 as shown in Figure 8 3 The area of the triangle that has these three points as vertices is 0 l i ap 2 oi k KE l 2 2 x A i 3 l 4 4 3 4 3 0 1 2 1 2 0 This result is generalized as follows Test for Collinear Points Three points x Yih 5 Y and xX y are collinear lie on the same line if and only if m Hy G Ja Q MEES o S Copyright 2010 Cengage Learning All Rights Reserved May not be copied sca
83. the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it B REVIEW EXERCISES In Exercises 1 and 2 form the augmented matrix for the system of linear equations Review Exercises 649 In Exercises 11 22 use matrices and elementary row operations to solve if possible the system of l 3x 10y 15 2 8x 7y 4z 12 5x 4y 22 3x 5y 2z 20 5x 3y 32 26 equations 11 5x 4y zip 2 i t 12 2 Sy 3x Ty 2 l In Exercises 3 and 4 write the system of linear equa tions represented by the augmented matrix Use vari ables x y z and w Be o My s a Wl 2B 8 oe 10 ge Be wy is 2h oF 3 2 41 1 Sr g 53 12 A 1 4 3 In Exercises 5 and 6 write the matrix in reduced row echelon form Coto l l x elt a 3 Ble a gt 2 2 S wai t t In Exercises 7 10 use the matrix capabilities of a graphing utility to write the matrix in reduced row echelon form 3 8 Ala og 0 ols Og ide ih ee i O Soo Dh op h De oh a in guig 4 8 16 A Pp a wi S f 3 eb Bh F 10 3 ti Il I 13 2x y gr y SS 14 0 2x O 1y 0 4x 0 5y 0 01 Ib 2x yt2z 4 16 2x 3y z 10 2x 2y 5 24 S 32 22 yp 62 12 4g 2y 3z 2 17 4 4y 4z 5 18 2x 3y 32 3 4x 2y Bz 1 Sx 3y 8z 6 19 x yt 2z
84. the second row to obtain a 0 below the leading 1 What effect does this operation have on the graph of the corresponding linear equation Graph of the system 2x 4y 9 x 5y 15 e Each time the 2 x 3 matrix is transformed the graph of the correspond ing linear equations is displayed What do you notice about the point of intersection each time Questions for Further Exploration 1 Is finding a point of intersection using the program 3 Run the program using the following linear system more or less accurate than finding the point of a 2 i i a E intersection using the zoom and trace features Explain your reasoning and give an example 2x y 3 2 Run the program to find the solution to the follow Describe what happens and why ing linear system 4 A system of equations with three variables has a dy 5 corresponding 3 x 4 augmented matrix Write a program that will transform a 3 x 4 matrix into reduced row echelon form At the end of the pro Why is only one line drawn in all but the last gram display the final matrix screen Verify the program s solution by hand 3x Sy 3 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience
85. tity matrix to the coefficient matrix to obtain A I l 4 l 0 i 4 Oo ty Then applying Gauss Jordan elimination to this matrix you can solve both systems with a single elimination process as follows 4 l 0 p en of T sW Sr a Me og RRs e r f t 1 tnd J Thus from the doubly augmented matrix A Z you obtained the matrix J A Tt A I AT a S I 10 gt wW ff 2 a y aI s oO 4 ih we This procedure or algorithm works for any square matrix that has an inverse Finding an Inverse Matrix Let A be a square matrix of order n 1 Write the n x 2m matrix that consists of the given matrix A on the left and the n x n identity matrix on the right to obtain A Z Note that we separate the matrices A and by a dotted line We call this process adjoining the matrices A and 2 If possible row reduce A to using elementary row operations on the entire matrix A 7 The result will be the matrix 7 A If this is not possible A is not invertible Check your work by multiplying to see that AA I A A ta Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to re
86. ue of 4A Give the reason for your dns wWET Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Chapter Project 653 CHAPTER PROJECT Row Operations and Graphing In this project you will investigate the graphical interpretation of elemen tary row operations a Solve the following systems by hand using Gauss Jordan elimination 2x 4y 9 6x 2y I9 x Sy 15 3xx y b Enter the row operations program listed in the appendix into a graphing calculator This program demonstrates how elementary row operations used in Gauss Jordan elimination may be depicted graphically For each system in part a run the program using a 2 x 3 matrix that corre sponds to the system of equations Compare the results of the program with those you obtained in part a c During the running of the program a row of the matrix is multiplied by a constant What effect does this operation have on the graph of the corresponding linear equation d During the running of the program a multiple of the first row of the matrix is added to
87. ust produce zeros above each of the leading 1 s as follows u 9 19 Second ohimni Nas v ni 0 l 3 5 Aboye 4s lemline 2 ww FR 7 0 si Muru coltre Ia ki ve tS Leela 0 2 Now converting back to a system of linear equations you have x y z 2 The beauty of Gauss Jordan elimination is that from the reduced row echelon form you can simply read the solution Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 598 8 J Matrices and Determinants The elimination procedures described in this section employ an algorithmic approach that ts easily adapted to computer use However the procedure makes no effort to avoid fractional coefficients For instance if the system given in Example 7 had been listed as 2x Sy 5z 17 X 2y r32c 9 x By 4 the procedure would have required multiplication of the first row by a which would have introduced fractions in the first row For hand computations fractions can sometimes be avoided by judiciously choosing the order in
88. ut of the matrix 3 we e L a d Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it LA P val 608 amp 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 A B and C be m x n matrices and let c and d be scalars lLA B B A Commutative Property of Matrix Addition 2A B C A B C Associative Property of Matrix Addition 3 cd A c dA Associative Property of Scalar Multiplication 4 IASA Scalar Identity ai c A B cA cB Distributive Property 6 c F dA cA dA Distributive Property Note that the Associative Property of Matrix Addition allows you to write expressions such as A B C without ambiguity because the same sum occurs no matter how the matrices are grouped In other words you obtain the same sum whether you group A B Cas A B CorasA B C This same reasoning applies to sums of four or more matrices EXAMPLE 4 Add
89. ve additional content at any time if subsequent rights restrictions require it In Exercises 51 58 evaluate the determinant Use a graphing utility lo verify your result 50 30 S1 T 10 8 53 54 ieai t Oo 2 SS od iG 56 0 d 3 0 4 O GAayti 57 58 ae ae ae 0 3 4 1 Ww Oo tn 6 4 6 2 3 In Exercises 59 66 use a graphing utility to solve if possible the system of linear equations using the inverse of the coefficient matrix Ss SPSS 3x 4y 5 60 x 3y 23 x 2y 18 a i aa a 2 br gaam 4y 3y 4z 62 x i dai Aa 8 2x4 Ty 32 19 s o32 3 63 xt 3 taS 2 2h SYS Z 1 2x 4y 2 64 2x 4y Z 3x 4y 2z 14 j Prze 6 Ys 4x 3y 2 yt3z 66 3r 3y 4z y yr 2z 4x 7v 102 OO Nn iI om 651 Review Exercises In Exercises 67 70 use Cramer s Rule to solve if pos sible the system of equations 67 69 x 2y 5 68 2x y 10 x y 3x 2y 20x B8y 11 70 13x 6v 17 l2x 24y 21 26x 12y 8 In Exercises 71 74 use a graphing utility and Cramer s Rule to solve if possible the system of equa tions 7L 34 6y 5 6x I4y ll 72 O0 4x 0 8y 1 6 0 2x 0 3y 2 2 Weak 3y B 2 kp ey Ss 4 we y be 4 74 lx 21ly Tz 10 4x Dy 2z2 4 564 2ly fe 3 75 Mixture Problem A flor
90. wo columns of A then B A 3 4 4 3 a 2 5h 3 2 6 l 2 6 2 3 4 l 6 2 b 2 2 0 2 2 0 pu Z 3 4 72 If A and B are square matrices and 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 A to another column of A then B A 4 to A ai a0 ee ac Ss 4 2 Ht 10 6 b 2 3 4 2 3 4 z G amp 3 I ve 2 IA and B are square matrices and B is obtained from A by multiplying a row of A by a nonzero constant e or multiplying a column of A by a nonzero constant e then B cA a b 5 p lt h th 10 3 7 S 12 4 15 4 3 hh e ft 1d ee a u w u Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Figure 8 1 aS ik 8 5 Applications of Matrices and Determinants 637 Applications of Matrices and Determinants Area of a Triangle Lines in the Plane Cramer s Rule Cryptography Area of a Triangle In this section
91. you will study some additional applications of matrices and determinants The first involves a formula for finding the area of a triangle whose vertices are given by three points on a rectangular coordinate system The area of a triangle with vertices x y gt Y2 and xz y3 is given by OY Area a 3 Ya Xa Ya 1 where the symbol indicates that the appropriate sign should be chosen to yield a positive area EXAMPLE 1 4 Finding the Area of a Triangle Find the area of a triangle whose vertices are 1 0 2 2 and 4 3 as shown in Figure 8 1 Solution Let xp y 1 0 x y2 2 2 and x3 y3 4 3 Then to find the area of a triangle evaluate the determinant x J l l 0 l r Wy HEA 2 2 l X Wy l 4 A 2 l 2 2 2 api a z 3 i ay 4 l ti 4 3 0 1 2 3 Using this value you can conclude that the area of the triangle is l 0 l 3 Area 2 2 I 3 2 2 4 3 l Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 638 8 J Matrices and De

x - The405

Contents

Download Pdf Manuals

Related Search

Related Contents