INVERSE OF A MATRIX AND ITS APPLICATION O Q

Transcription

1 Inverse f A tri And Its Appliction DUE - III IVERSE F A ATRIX AD ITS AICATI ET US CSIDER A EXAE: Abhinv spends Rs. in buying pens nd note books wheres Shntnu spends Rs. in buying 4 pens nd note books.we will try to find the cost of one pen nd the cost of one note book using mtrices. et the cost of pen be Rs. nd the cost of note book be Rs. y. Then the bove informtion cn be written in mtri form s: 4 y This cn be written s A X where A 4, X y ur im is to find X y B nd B In order to find X, we need to find mtri A so tht X A B This mtri A - is clled the inverse of the mtri A. In this lesson, we will try to find the eistence of such mtrices.we will lso lern to solve system of liner equtions using mtri method. ATHEATICS 47

2 DUE - III BJECTIVES Inverse f A tri And Its Appliction After studying this lesson, you will be ble to : define minor nd cofctor of n element of mtri; find minor nd cofctor of n element of mtri; find the djoint of mtri; define nd identify singulr nd non-singulr mtrices; find the inverse of mtri, if iteists; represent system of liner equtions in the mtri formax B; nd solve system of liner equtions by mtri method. EXECTED BACKGRUD KWEDGE Concept of determinnt. Determinnt of mtri. tri with its determinnt of vlue. Trnspose of mtri. inors nd Cofctors of n element of mtri.. DETERIAT F A SUARE ATRIX We hve lredy lernt tht with ech squre mtri, determinnt is ssocited. For ny given mtri, sy A 4 its determinnt will be. It is denoted by A. 4 Similrly, for the mtri A A , the corresponding determinnt is 48 ATHEATICS

3 Inverse f A tri And Its Appliction A squre mtri A is sid to be singulr if its determinnt is zero, i.e. A A squre mtri A is sid to be non-singulr if its determinnt is non-zero, i.e. A DUE - III Emple. Determine whether mtri A is singulr or non-singulr where () A 6 4 A 4 Solution: () Here, A 6 4 ( 6)( ) ( 4)( ) + Therefore, the given mtri A is singulr mtri. Here, A Therefore, the given mtri is non-singulr. Emple. Find the vlue of for which the following mtri is singulr: A ATHEATICS 49

4 DUE - III Solution: Here, Inverse f A tri And Its Appliction A + + ( 6 ) + ( ) + ( ) Since the mtri A is singulr, we hve A A 8 8 or Thus, the required vlue of is. Emple. Given A Solution: Here, A 6 trnspose of the mtri. 6 This gives A 6. Show tht A A, where A denotes the ow, A () nd A' () 6 From () nd (), we find tht A A. IRS AD CFACTRS F THE EEETS F SUARE ATRIX Consider mtri A ATHEATICS

5 Inverse f A tri And Its Appliction The determinnt of the mtri obtined by deleting theith row nd jth column of A, is clled the minor of ij nd is denotes by ij. Cofctor C ij of ij is defined s DUE - III C ij i j ( ) + For emple, inor of ij nd C Cofctor of ( ) + ( ) Emple.4 Find the minors nd the cofctors of the elements of mtri A Solution: For mtri A, A (minor of ) ; C ( ) ( ) + (minor of ) 6; C ( ) ( ) 6 + (minor of 6) ; C ( ) ( ) + 4 (minor of ) ; C ( ) ( ) Emple. Find the minors nd the cofctors of the elements of mtri A ATHEATICS

6 DUE - III Inverse f A tri And Its Appliction Solution: Here, + 7 C + ; ( ) C + ; ( ) ; C ( ) ; C ( ) 6 + ( 4) 7; C ( ) ( ) ; C ( ) ( 6 ) 6; C ( ) 6 6 C + ( ) ; ( ) nd + ( 6) ; C ( ) CHECK YUR RGRESS.. Find the vlue of the determinnt of following mtrices: () A B ATHEATICS

7 Inverse f A tri And Its Appliction. Determine whether the following mtri re singulr or non-singulr. () A 9 6. Find the minors of the following mtrices: () A B 6 4. () Find the minors of the elements of the nd row of mtri A 4 Find the minors of the elements of the rd row of mtri A 4. Find the cofctors of the elements of ech the following mtrices: () A B 6 6. () Find the cofctors of elements of the nd row of mtri A 4 Find the cofctors of the elements of the st row of mtri A 6 4 DUE - III ATHEATICS

8 f A Inverse f A tri And Its Appliction DUE - III 7. I 4 nd B 7 4, verify tht () A A nd B B AB A B BA. ADJIT F A SUARE ATRIX et A 7 be mtri. Then A 7 et ij nd C ij be the minor nd cofctor of ij respectively. Then + 7 7; C ( ) ; C ( ) + ; C ( ) + ; C ( ) We replce ech element of A by its cofctor nd get B 7...( ) The trnspose of the mtri B of cofctors obtined in () bove is B 7...( ) The mtri B obtined bove is clled the djoint of mtri A. It is denoted by Adj A. Thus, djoint of given mtri is the trnspose of the mtri whose elements re the cofctors of the elements of the given mtri. Working Rule: To find the Adj A of mtri A: () replce ech element of A by its cofctor nd obtin the mtri of cofctors; nd tke the trnspose of the mri of cofctors, obtined in (). 4 ATHEATICS

9 Inverse f A tri And Its Appliction Emple.6 Find the djoint of A 4 DUE - III Solution: Here, A 4 et A be the cofctor of the element. ij ij + + Then, A ( ) ( ) A ( ) ( ) + A + ( ) ( ) A ( ) ( 4) 4 We replce ech element of A by its cofctor to obtin its mtri of coftors s 4 Trnspose of mtri in () is AdjA. Thus, Adj A 4...() Emple.7 Find the djoint of A Solution: Here, 4 A 4 et A ij be the cofctor of the element ij of A 4 + Then A ( ) 4 6 ( ) + ; A ( ) ( ) 4 + ( ) ( 6 ) 6 ; A ( ) 4 ( ) + A ATHEATICS

10 DUE - III + A Inverse f A tri And Its Appliction ( ) ( ) + ; A ( ) ( + ) 7 + ( ) ( 8) 9 ; A 4 ( ) ( + 6) 7 + A nd A ( ) ( ) Replcing the elements of A by their cofctors, we get the mtri of cofctors s Thus, Adj A Verifiction: If A is ny squre mtri of order n, then A(Adj A) (Adj A) A A I n where I n is the unit mtri of order n. 4 () Consider A 4 Then A or A ( ) (4) Here, A ; A ; A 4 nd A Therefore, Adj A ow, A (AdjA) A I 7 () Consider, A Then, A ( 6 ) (4) + 7 (+) Here, A 7; A ; A 6 ATHEATICS

11 Inverse f A tri And Its Appliction A ; A ; A Therefore, Adj A ow (A) (Adj A) A 6; A ; A Also, (Adj A) A 9 7 ( ) (_) 6 9 A I 7 A I DUE - III ote : If A is singulr mtri, i.e. A, then A (Adj A) CHECK YUR RGRESS.. Find djoint of the following mtrices: () 6. Find djoint of the following mtrices : () i i i Also verify in ech cse tht A(Adj A) (Adj A) A A I. i c b d (c) cos sin sin cos ATHEATICS 7

12 DUE - III Inverse f A tri And Its Appliction. Verify tht A(Adj A) (Adj A) A A I, where A is given by () (c) cos sin sin cos (d).4 IVERSE F A ATRIX Consider mtri A i.e., or B c u b d c b. We will find, if possible, mtri d y such tht AB BA I v u y v + bu y + bv c + du cy + dv n compring both sides, we get + bu y + bv c + du cy + dv Solving for, y, u nd v, we get provided d d, y d bc bc, i.e., b, u d bc c b d c, v d bc d bc 8 ATHEATICS

13 Inverse f A tri And Its Appliction Thus, or B d d bc c d bc B d bc d c b d bc d bc b It my be verified tht BA I. It my be noted from bove tht, we hve been ble to find mtri. B d bc d c b A Adj A...() This mtri B, is clled the inverse of A nd is denoted by A -. For given mtri A, if there eists mtri B such tht AB BA I, then B is clled the multiplictive inverse of A. We write this s B A -. DUE - III ote: bserve tht if d bc, i.e., A, the R.H.S. of () does not eist nd B (A - ) is not defined. This is the reson why we need the mtri A to be non-singulr in order tht A possesses multiplictive inverse. Hence only non-singulr mtrices possess multiplictive inverse. AlsoB is non-singulr ndab -. Emple.8 Find the inverse of the mtri A Solution : A 4 4 Therefore, A A is non-singulr. It mens A hs n inverse. i.e. A - eists. ow, Adj A A 4 dj A A 4 ATHEATICS 9

14 DUE - III Inverse f A tri And Its Appliction ote : Verify tht AA - A - A I Emple.9 Find the inverse of mtri A Solution : Here, A A ( 4) ( ) (4 + ) ( 9) ( ) (9) A eists. et A ij be the cofctor of the element ij. Then, 6 ( ) 4 9 4, + A 6 ( ) ( ). + A ( ) 4 9, 4 + A ( ) 8 4 ( + ) + A ( ) +, + A ( ) ( ) 4 + A 6 ATHEATICS

15 Inverse f A tri And Its Appliction ( ), 6 + A DUE - III ( ) ( 8 + ) 6 + A + nd A ( ) tri of cofctors 9 9. Hence AdjA A A Adj A ote : Verify tht A - A AA - I Emple. If A Solution : (i) Here, AB nd B ; find (i) (AB) (ii) B A (iii) Is (AB) B A? AB Thus, ()AB eists. et us denote AB by Cij ATHEATICS 6

16 o w, B DUE - III Inverse f A tri And Its Appliction et C ij be the cofctor of the element c ij of C. Then, C ( ) + () C ( ) + () Hence, Adj (C) C ( ) + ( 4) 4 4 C ( ) + ( ) C Adj ( C) C 4 C (AB) (ii) To find B A, first we will find B -. B - eists. et Bij be the cofctor of the element bij of B then B ( ) + ( ) B ( ) + () Hence, Adj B B () + () nd B. B AdjB Also, A B ( ) + ( ) Therefore, A Therefore, A - eists. et Aij be the cofctor of the element ij of A then A ( ) + () A ( ) + () A () + () nd A ( ) + () 6 ATHEATICS

17 Inverse f A tri And Its Appliction Hence, Adj A A A A Adj DUE - III Thus, B A + + (iii) From (i) nd (ii), we find tht (AB) B A Hecne, (AB) B A CH ECK YUR RGRESS.. Find, if possible, the inverse of ech of the following mtrices: () 4 (c). Find, if possible, the inverse of ech of the following mtrices : () 4 4 Verify tht A A AA I for () nd. ATHEATICS 6

18 DUE - III. If A 4 4. Find ( A ) if A. If A nd B nd B 4 Inverse f A tri And Its Appliction 4 b + c c b c b c + b b c c + b, verify tht ( ) AB B A show tht ABA is digonl mtri. 6. If φ( ) 7. If A 8. If A 9. If A. If A c cos sin tn b + tn bc sin cos show tht, A A, show tht A A, show tht ( ) ( ). cos sin sin cos, show tht A ( + bc + ) I A, show tht A A 64 ATHEATICS

19 Inverse f A tri And Its Appliction. SUTI F A SYSTE F IEAR EUATIS. In erlier clsses, you hve lernt how to solve liner equtions in two or three unknowns (simultneous equtions). In solving such systems of equtions, you used the process of elimintion of vribles. When the number of vribles invovled is lrge, such elimintion process becomes tedious. You hve lredy lernt n lterntive method, clled Crmer s Rule for solving such systems of liner equtions. We will now illustrte nother method clled the mtri method, which cn be used to solve the system of equtions in lrge number of unknowns. For simplicity the illustrtions will be for system of equtions in two or three unknowns... ATRIX ETHD In this method, we first epress the given system of eqution in the mtri formax B, where A is clled the co-efficient mtri. For emple, if the given system of eqution is + b y c nd + b y c, we epress them in the mtri eqution form s : b c b y c b Here, A, X b y c nd B c If the given system of equtions is + b y + c z d nd + b y + c z d nd + b y + c z d,then this system is epressed in the mtri eqution form s: Where, A b c b c b c y z b c b c b c d d d, X y z nd B Before proceding to find the solution, we check whether the coefficient mtria is non-singulr or not. d d d DUE - III ATHEATICS 6

20 DUE - III Inverse f A tri And Its Appliction ote: If A is singulr, then A. Hence, A - does not eist nd so, this method does not work. Consider eqution AX B, where A b, X b y c nd B c When A, i.e. when b _ b, we multiply the eqution AX B with A - on both side nd get A - (AX) A - B (A - A) X A - B IX A - B ( A - A I) Since A X A - B b b b b b b X b b y b b b c c + c c b c c b c b c b b c + c b b, we get b c Hence, b b c b c nd y b c b Emple. Using mtri method, solve the given system of liner equtions. 4 y...(i) + 7y W Solution: This system cn be epressed in the mtri eqution form s 4 7 U V y...(ii) 66 ATHEATICS

21 Inverse f A tri And Its Appliction Here, A so, 4, X 7 (ii) reduces to AX B ow, A y, nd B...(iii) DUE - III Since A, A - eists. ow, on multiplying the eqution AX B with A - on both sides, we get i.e. A - (AX) A - B (A - A)X A - B IX A - B X A - B Hence, X ( Adj A ) B A or So, y y y , y is unique solution of the system of equtions. Emple. Solve the following system of equtions, using mtri method. y 7 + y Solution : The given system of equtions in the mtri eqution form, is 7 y ATHEATICS 67

22 DUE - III Inverse f A tri And Its Appliction or, AX B...(i) where A ow, A, X A - eists Since, Adj (A) A From (i), we hve X A B y 7 nd B ( ) dj( A) A (ii) or, X or, 7 y 7 Thus, 7, y is the solution of this system of equtions. 7 Emple. Solve the following system of equtions, using mtri method. + y + z 4 y + z + y z U V W 68 ATHEATICS

23 Inverse f A tri And Its Appliction Solution : The given equtions epressed in the mtri eqution form s : y z 4 which is in the form AX B, where A, X y z nd B 4... (i) X A - B... (ii) Here, A () ( ) + (+4) 6 A - eists. DUE - III 8 Also, Adj A Hence, from (ii), we hve X A B A AdjA. B or, X y z ATHEATICS 69

24 DUE - III Inverse f A tri And Its Appliction Thus,, y nd z is the solution of the given system of equtions. Emple.4 Solve the following system of equtions, using mtri method : +y+z y+z +y z Solution: The given system of eqution cn be represented in the mtri eqution form s : y z i.e., AX B () where A ow, Hence, A A - eists. Also, Adj A From i.e., A, X A Adj A (i) we hve X A - B y z 9 y z nd B ( 9) ( ) + (6 +) ATHEATICS

25 Inverse f A tri And Its Appliction DUE - III so, 9, y 4 9, z 9 is the solution of the given system..6 CRITERI FR CSISTECY F A SYSTE F EUATIS et AX B be system of two or three liner equtions. Then, we hve the following criteri : () If A, then the system of equtions is consistent nd hs unique solution, given byxa - B. () If A, then the system my or my not be consistent nd if consistent, it does not hve unique solution. If in ddition, () (Adj A) B, then the system is inconsistent. (Adj A) B, then the system is consistent nd hs infinitely mny solutions. ote : These criteri re true for system of 'n' equtions in 'n' vribles s well. We now, verify these with the help of the emples nd find their solutions wherever possible. () + 7y y This system is consistent nd hs unique solution, becuse 7 eqution is y i.e. AX B... (i) where, A 7, X y nd B 7 Here, the mtri ATHEATICS 7

26 DUE - III Here, A ( ) nd A A Adj A 7 9 From (i), we hve X A B i.e., y Inverse f A tri And Its Appliction... (ii) [From (i) nd (ii)] Thus, 4, nd y 9 9 is the unique solution of the given system of equtions. + y y 8 This system is incosisntent i.e. it hs no solution becuse 6 4 In the mtri form the system cn be written s or, where A y AX B, X 6 4 y 7 nd B 8 Here, A 4 6 Adj A Also, (Adj A) B ATHEATICS

27 Inverse f A tri And Its Appliction Thus, the given system of equtions is inconsistent. (c) y 7 9 y U V W mny solutions, becuse 9 This system is consistent nd hs infinitely 7 In the mtri form the system cn be written s or, 7 9 y AX B, where A ; X 9 y 7 nd B DUE - III Here, A 9 () 9 () Adj A Also, (Adj A)B 7 9 The given system hs n infinite number of solutions. Till now, we hve seen tht (i) (ii) (iii) (iv) if A nd (Adj A) B, then the system of equtions hs non-zero unique solution. if A nd (Adj A) B, then the system of equtions hs only trivil solution y z if A nd (Adj A) B, then the system of equtions hs infinitely mny solutions. if A nd (Adj A) B, then the system of equtions is inconsistent. ATHEATICS 7

28 DUE - III Inverse f A tri And Its Appliction et us now consider nother system of liner equtions, where A nd (Adj A) B. Consider the following system of equtions + y + z + y + z y + z 6 In mtri eqution form, the bove system of equtions cn be written s i.e., AX B where A y z 6, X y z nd B 6 ow, A Also, (Adj A ) B ( C C ) Since A nd (Adj A ) B, y A Adj A B z ( ) 6 [Verify (Adj A) yourself] 74 ATHEATICS

29 Inverse f A tri And Its Appliction 8 8 which is undefined. The given system of liner eqution will hve no solution. Thus, we find tht if A nd (Adj A) B then the system of equtions will hve no solution. We cn summrise the bove finding s: DUE - III (i) (ii) (iii) (iv) If A nd (AdjA) B then the system of equtions will hve non-zero, unique solution. If A nd (Adj A) B, then the system of equtions will hve trivil solutions. If A nd (AdjA) B, then the system of equtions will hve infinitely mny solutions. If A nd (AdjA) B, then the system of equtions will hve no solution Inconsistent. Emple. Use mtri inversion method to solve the system of equtions: (i) 6 + 4y 9 + 6y (ii) y + z + y z y z Solution: (i) i.e., The given system in the mtri eqution form is y AX B where, A X 6 4 y 9 6, nd B 6 4 ow, A The system hs either infinitely solutions or no solution. ATHEATICS 7

30 DUE - III Inverse f A tri And Its Appliction k et k, then 6k + 4y gives y utting these vlues of nd y in the second eqution, we hve F 9 6 ki k + HG K J 8k + 6 8k 6 (ii) 6 6, which is true. The given system hs infinitely mny solutions. These re k k, y, where k is ny rbitrry number. The given equtions re ( ) ( ) ( ) y + z + y z y z In mtri eqution form, the given system of equtions is y z i.e., AX B A, X y where, nd B z ow, A + + ( +)+( )+( ) + 76 ATHEATICS

31 Inverse f A tri And Its Appliction The system hs either infinitely mny solutions or no solution et z k. Then from (), we hve y k; nd from (), we hve +y +k ow, we hve system of two equtions, nmely DUE - III y k + y + k y + y k et A X, nd B k k + k Then A 4 + Here, A eists. A Adj A A The solution is X A B 4 k + k k + k + 4 k +, y k +, where k is ny number. utting these vlues of, y nd z in (), we get F I k + HG K J ( k) k + k, which is true. ATHEATICS 77

32 DUE - III Inverse f A tri And Its Appliction The given system of equtions hs infinitely mny solutions, given by 4 k +, y k + nd z k, where k is ny number..7 HGEEUS SYSTE F EUATIS A system of liner equtions AX B with mtri, B, null mtri, is clled homogeneous system of equtions. Following re some systems of homogeneous equtions: (i) + y + y (ii) + y z y + z y 6z (iii) + y z y + z y z et us now solve system of equtions mentioned in (ii). Given system is + y z y + z y 6z In mtri eqution form, the system (ii) cn be written s 6 y z i.e., where AX A, X y nd B 6 z ow A ( + ) ( 6 ) ( + 6) ATHEATICS

33 Inverse f A tri And Its Appliction But B (Adj A) B. Thus, y ( Adj A) B A z., y nd z. i.e., the system of equtions will hve trivil solution. Remrks: For homogeneous system of liner equtions, if A nd (Adj A) B. There will be only trivil solution. ow, cousider the system of equtions mentioned in (iii): DUE - III + y z y + z y z In mtri eqution form, the bove system (iii) cn be written s i.e., AX where, A ow, A Also, B y z (Adj A) B, X y nd B z ( 4 + ) ( ) ( + 6) + ATHEATICS 79

34 DUE - III Thus, y z A (Adj A) B Inverse f A tri And Its Appliction The system of equtions will hve infinitely mny solutions which will be non-trivil. Considering the first two equtions, we get + y z y z Solving, we get z, y z, let z k, where k is ny number. Then k, y k nd z k re the solutions of this system. For system of homogenous equtions, if A nd, (AdjA) B, there will be infinitely mny solutions. CHECK YUR RGRESS.4. Solve the following system of equtions, using the mtri inversion method: () + y 4 + y 7 y 7y (c) + 4y (d) y + 6 y y 8. Solve the following system of equtions using mtri inversion method: () + y + z + y + z y + z + y z + y z 7 + y + z (c) + y + z (d) + y z y + z + y z + y + z y z 8 ATHEATICS

35 Inverse f A tri And Its Appliction. Solve the following system of equtions, using mtri inversion method: () + y + z y + z (c) + y + DUE - III y + z + y z y + z y + z 4 y + z z + 4. Determine whether the following system of equtions re consistent or not. If consistent, find the solution: () y y + y 7 4 6y (c) + y + z (d) + y z y z 7 4 y + z + y + z + y 4z ET US SU U A squre mtri is sid to be non-singulr if its corresponding determinnt is non-zero. The determinnt of the mtri A obtined by deleting the i th row nd j th column of A, is clled the minor of ij. It is usully denoted by ij. The cofctor of ij is defined s C ij () i+j ij Adjoint of mtri A is the trnspose of the mtri whose elements re the cofctors of the elements of the determint of given mtri. It is usully denoted by AdjA. If A is ny squre mtri of order n, then A (Adj A) (Adj A) A A I n where I n is the unit mtri of order n. For given non-singulr squre mtri A, if there eists non-singulr squre mtri B such tht AB BA I, then B is clled the multiplictive inverse of A. It is written s B A. nly non-singulr squre mtrices hve multiplictive inverse. If + b y c nd + b y c, then we cn epress the system in the mtri eqution form s ATHEATICS 8

36 DUE - III Thus, i f A b c b y c b X b y, nd B b b X A B b b Inverse f A tri And Its Appliction c c, c c then A system of equtions, given by AX B, is sid to be consistent nd hs unique solution, if A. A system of equtions, given by AX B, is sid to be inconsistent, if A nd (Adj A) B. A system of equtions, given by AX B, is sid be be consistent nd hs infinitely mny solutions, if A, nd (AdjA) B. A system of equtions, given by AX B, is sid to be homogenous, if B is the null mtri. A homogenous system of liner equtions, AX hs only trivil solution... n, if A A homogenous system of liner equtions, AX hs infinitely mny solutions, if A SURTIVE WEB SITES TERIA EXERCISE. Find A, if () A 4 4 A 7 8 ATHEATICS

37 Inverse f A tri And Its Appliction. Find the djoint of A, if () A 7 4 A Also, verify tht A(Adj A) A I (Adj A) A, for () nd. Find A, if eists, when () 6 7 Also, verify tht (A ) (A ), for (), nd (c) 4. Find the inverse of the mtri A, if () A (c) A 4. Solve, using mtri inversion method, the following systems of liner equtions DUE - III () + y 4 + y y 9 + 6y (c) + y + z y z y z 9 (d) y + z 4 + y z + y + z (e) + y z y + z + y z 6. Solve, using mtri inversion method ; + ; + y z y z y z ATHEATICS 8

38 DUE - III Inverse f A tri And Its Appliction 7. Find non-trivil solution of the following system of liner equtions: + y + 7z 4 y z + 9y + z 8. Solve the following homogeneous equtions : + y z + y z () y + z + 6y z + y z + 4y 9z 9. Find the vlue of p for which the equtions + y + z p + y + z py + y + z pz hve non-trivil solution. Find the vlue of for which the following system of eqution becomes consistent y + 4 y 8y + 84 ATHEATICS

39 Inverse f A tri And Its Appliction ASWERS CHECK YUR RGRESS.. (). () singulr non-singulr. () 4; 7; ; DUE - III ; ; 6; 4. () ; 7; ; ;. () C 7; C 9; C ; C C 6; C ; C 4; C 6. () C ; C 8; C C 6; C ; C CHECK YUR RGRESS.. (). () 6 d c b i i i i (c) cos sin sin cos CHECK YUR RGRESS.. (). () (c) ATHEATICS 8

40 DUE - III 4. (A ) Inverse f A tri And Its Appliction CHECK YUR RGRESS.4. () y 6, 7 7 6, y 7 (c), y (d) 7, y 8. (), y, z, y, z (c), y, z (d), y, z. (), y, z, y, z (c), y, z 6 4. () Consistent;, y (c) 9 Consistent; infinitely mny solutions Inconsistent (d) Trivil solution, y z 86 ATHEATICS

41 ATHEATICS 87 DUE - III Inverse f A tri And Its Appliction TERIA EXERCISE. () 4. () () (c) ()

42 DUE - III. (), y k, y k (c), y, z (d), y, z Inverse f A tri And Its Appliction (e), y, z 6., y, z 7. k, y k, z k 8. () k, y k, z k y z k 9. p,, ATHEATICS