M.A. (ECONOMICS) PART-I PAPER - III BASIC QUANTITATIVE METHODS

Transcription

1 M.A. (ECONOMICS) PART-I BASIC QUANTITATIVE METHODS LESSON NO. 9 AUTHOR : SH. C.S. AGGARWAL MATRICES Mtrix lger eles oe to solve or hdle lrge system of simulteous equtios. Mtrices provide compct wy of writig equtio system eve extremely lrge oe. It is pplicle oly i lier equtio system. Ifct, it hs the lterte me lier lger. Lierity ssumptio frequetly dopted i ecoomics my i certi cse e quite resole d justified. O this sis, Mtrix lger is developed. x + y z x y + z The coefficiets of the ove two equtios c e writte s: A The ove rectgulr rry is clled mtrix. The ove mtrix A cotis rows (horizotl lies) d colums (verticl lies). It c e sid tht A is mtrix of order d is red s y mtrix. Defiitio : A mtrix is rrgemet of m ordered umers cosistig of m rows d colums. This mtrix is clled mtrix of order m d is red s m y mtrix. Mtrices re deoted y cpitl letters A,B, etc. A mtrix of order mx is writte s : A m m ij j mj m m The s re clled elemets of mtrix A. The ove mtrix A hs m rows d colums. A mtrix is rectgulr rry of umers d is eclosed y rckets or with doule verticl lies. The elemets i the ith row d jth colum is deoted y ij. Briefly we write: A [ ij ] Thus the mtrices of order, d re writte s :

2 Equlity of Mtrices : Two mtrices re equl if d oly if they hve the sme umer of rows d the sme umer of colums d the correspodig elemets i the two re equl e.g. A c d d B w t s u Now A B if w s c t d u Symoliclly if A [ ij ], [ ij ] The A B, if A d B hve sme dimesios d ij ij for ech (i, j) Types of Mtrices : Squre Mtrix d Rectgulr Mtrix : If m, i.e. if the umer of rows re equl to the umer of colums, we hve squre mtrix, e.g. A c d d B d g e h c f i A mtrix which is ot squre mtrix is clled rectgulr mtrix, e.g. A d B re rectgulr mtrices of order d respectively. Digol Mtrix : It is squre mtrix i which ll the elemets re zero, except those i the ledig digol, e.g. A d d it is digol Mtrix d Sclr Mtrix : A digol mtrix i which ll the digol elemets re equl is clled sclr mtrix. For exmple : A d d is Sclr Mtrix d Uit Mtrix : A sclr mtrix, ech of whose digol elemets is uity, is clled uit mtrix d is deoted y. I is uit mtrix of order

3 Zero or Null Mtrix : A Mtrix with every elemet equl to zero is clled zero or ull mtrix. It my e squre or rectgulr : O or O [ ] re zero mtrix of order d respectively. Su-mtrix : A mtrix otied y deletig some rows d some colums of give mtrix A is clled su mtrix of A, e.g. Suppose A d B is su mtrix of A d is otied y elimitig d row d d d th colums of mtrix A. Row d Colum Mtrix : A row mtrix hs oly oe row d is of order. Similrly colum mtrix hs oly oe colum d is of order m. Exmple : (i) A [ c] is row mtrix of order (ii) B is colum mtrix of order Upper d Lower Trigulr Mtrices : A squre mtrix ll of whose elemets elow the mi digol re zero is clled upper trigulr mtrix. If ll elemets ove the mi digol re zero the it is lower trigulr mtrix. For exmple : (i) A is upper trigulr mtrix of order (ii) B is lower trigulr mtrix of order Trspose of mtrix : The trspose of mtrix is otied y writig the ith row s the jth colum d the jth colum s the ith row. I other words, if we iterchge the rows d colums of m mtrix A, we get m mtrix A, which is clled the trspose of A, symoliclly. If A m ( ij ) the A m ( j i )

4 M.A. (ECONOMICS) PART - I For exmple if A 9 the A or A t 9 The symol ' or t deotes trspose If the mtrix is squre s i the ove exmple, the the trspose of the mtrix c e thought of s the mtrix with the sme mi digol, with ll the other elemets reflected i tht digol. If A the A Symmetric Mtrix : It is squre mtrix A such the ij ji i.e. (ij) th elemet of A is equl to (ji) th elmet of A. I other words A A where A is the trspose of mtrix A. For exmple : A c k t k t A c k t k t, B 9 9, B 9 9 A A B B A symmetric mtrix is reflectio; of itself i the mi digol. Skew Symmetric Mtrix : It is squre mtrix A i which ij - ji for ll vlues of i d j i other words (ij) th elemet of A is equl to the egtive of (ji)th elemet of A. Now ij - ji is skew symmetric mtrix or ij O j i Put ji ij ij O So ll the digol elemets of skew-symmetric mtrix A re zero. A k k o B, A k k o, B Thus A d B re two skew symmetric mtrices ecuse A A d B B' Idempotet mtrix : A squre mtrix A is sid to e idempotet if A A.A. (Uit mtrix is exmple of idempotet mtrix) Cojugte Mtrix : If A is m mtrix, the the m mtrix is otied y replcig ech elemet of A y its complex cojugte is clled Cojugte Mtrix of A d is deoted

5 y A. Thus if A ( ij ) the the mtrix A ij is clled the cojugte mtrix of A where ij is the complex cojugte of. Oviously A is rel if d oly if A A, i.e. if ll the elemets of A re rel. ij Hermiti Mtrix : A squre mtrix A is sid to e Hermiti Mtrix if e.g. the mtrix A i i i is Hermiti mtrix A A, Note : All the digol elemets of Hermiti mtrix re rel. Skew Hermiti Mtrix : A squre mtrix A is sid to e Skew Hermiti if A, e.g. the mtrix i i i i i i i i is skew Hermiti mtrix. Note : I skew Hermiti mtrix the digol elemets re purely imgiry or zero. Exmple : If A is y squre mtrix, show tht A+ A is Hermiti Sol. We kow tht squre mtrix A is Hermiti if A A Now (A + A ) A + (A ) A + A ((A ) A A + A Hece A A is Hermiti Orthogol Mtrix : A squre mtrix A is sid to the orthogol if A A A A Nilpotet Mtrix : A squre mtrix A is sid to e ilpotet of idex if A Exmple : A is ilpotet ecuse A Idempotet Mtrix : A mtrix reproduces itself whe multiplied y itself is clled idempotet mtrix. Tht is if A.A A the A is clled idempotet mtrix. For exmple, A is idempotet mtrix A.A A

6 M.A. (ECONOMICS) PART - I Exmple : Show tht A is idempotet mtrix. Solutio : Here A.A. 9 9 A Sice A.A. A Hece A is idempotet mtrix. Exmple : If A Prove tht B I -A(A'.A) - A. Show tht B is idempotet mtrix Solutio: Sice A A' A'.A d A A ' (-) dj (A'A) (A'A) - A'A dj (A'A)

7 M.A. (ECONOMICS) PART - I d A(A'.A) - A' 9 Now B I -A (A'.A) - A' (AA) -

8 Now B.B B B Sice B.B B B is idempotet mtrix Theorems o Idempotet. If AB A d BA B the A d B re idempotet Let us cosider the mtrix ABA Sice ABA (AB)A AA A Sice ABA (A) BA AB A Which gives A A Hece A is idempotet Agi BAB (BA) B B.B B BA B Also BAB B (AB) BA B ( AB A, BA B) Which gives B B Hece B is idempotet. If B is idempotet show tht A-B, is lso idempotet d tht ABBA o Sice B is idempotet B B Now A (-B) (-B). ( B) ( A B) B + B B + B ( B B d B B) B A

9 9 Sice A A A is idempotet Agi AB ( B) B.B B.B.B B d B.B B B B Similrly BA B (-B) B B.B B B Trce of mtrix I squre mtrix, ll those elemets ij for which i j (i.e. ij ) re clled the digol elemets d the lie log which they lie is clled the pricipl digol. The sum of the elemets i the pricipl digol of squre mtrix is clled the trce of the mtrix : e.g. if A the trce of A tr. A Sum of the digol elemets + + i ij ij m I geerl if A x x m Trce of A tr. A + + i ij To tke umericl exmple Let A The trce of A tr. A + + (-) Note : It my e oted tht trce of ull mtrix of y order is zero d trce of idetity mtrix of th order is For exmple :

10 Let O the tr. O + + Similrly let I The tr. I + + if we tke I the tr. I (upto ) Theorems o Trce. tr. (A') tr. A, A. eig squre mtrix ( ij ) of order.. tr. (AA') tr. (A'A). tr. (ka) K tr. (A), k eig sclr. tr. (A+B) tr. (A) + tr. (B) if A d B re squre mtrices of the sme order. tr. (AB) tr. (BA) if AB d BA re oth defied.. tr. (ABC) tr. (BCA) tr. (CAB), if ABC, BCA d CAB re ll defied.. Verifictio : tr. A' tr. A Let A the A' tr. A' + + tr. A. tr. (A.A') tr. (A'A) Let A

11 so tht A' d AA' tr. (AA') Similrly A'A tr. (A'A) tr. (AA') tr. (A'A). tr. (ka) k tr. (A) Let A The ka k k k k k k k k k tr. (ka) k + k + k K ( + + ) k tr. A. tr. (A+B) tr. (A) + tr. (B) Let A B

12 tr A + + tr B + + A + B + tr. (A+B) ( + ) + ( + ) + ( + ) ( + + ) + ( + + ) tr. (A) + tr. (B). tr. (AB) tr. (BA). Let A d B the A B tr. (AB) BA tr. (BA) ( ) tr. (AB) ( ) ( ) tr. (BA) Opertios of Mtrices Opertios of Mtrices iclude dditio, sutrctio d multiplictio of mtrices. They re discussed elow oe y oe. Additio of Mtrices : Two mtrices c e dded if d oly if they hve the sme dimetios (order). Whe this dimesiol requiremet is met, the mtrices re sid to e comfortmle for dditio. I such situtio, the sum of mtrix A ( ij ) d mtrix B ( ij ) is the dditio of ech pir of correspodig elemets. I other words, the sum of the m mtrix A d the (m ) mtrix B is sid to e mtrix C such tht c ij ij + ij d we cll the process of formig C the dditio of B to A.

13 M.A. (ECONOMICS) PART - I A d c B k t s r The the sum of these A + B k d t c s r Suppose tht A deotes the qutities ought y Mr. Alie i ech of the three weeks of four differet goods. Let B deote the qutities ought of the sme goods y Mr. Alle i three weeks. The totl purchses will e : A d B A + B It two mtrices hve differet dimesios/order, their dditio is ot defied, e.g. A d B Sice A is of order d B order hece their sum is ot possile. Sutrctio of Mtrices : The differece etwee two mtrices (sutrctio) c e kow if d oly if the two mtrices hve the sme dimesios. I such situtio the differece etwee the mtrices i.e. A-B c e otied y sutrcig the elemets of B from the correspodig elemets of A. i other words. (d ij ) ( ij ) ( ij ) it c lso e stted tht if A ( ij ) m, B ( ij ) m the A B ( ij ) m ( ij ) m for exmple : A d c B u t s r The A B u d t c s r

14 M.A. (ECONOMICS) PART - I or, if the followig iformtio is give to us A d B 9 The the differece etwee two mtrices A B 9 Thus the sutrctio opertio my e tke ltertively s dditio opertio ivolvig mtrix A d other mtrix (-) B, or A-B A + (-B). Multiplictio of Mtrices : The multiplictio of two mtrices is cotiget upo the stisfctio of other dimesiol requiremet. The coformility, coditios for the multiplictio is tht the colum dimesio of mtrix A (the led mtrix i the expressio AB) must e equl to the row dimesio of B (the lg ) mtrix) for exmple : Let A c c, B r q p Mtrix A hs colums d B hs rows. The procedure of Multiplictio is tht ech elemet of row from the first mtrix is to e multiplied y the correspodig elemet of colum of the secod mtrix d the sum of these product costitutes elemet of the mtrix AB. Thus the product of mtrix AB, i the ove metioed mtrices A d B will e: AB r c r c q p q p d it is mtrix. The procedure will e similr for higher dimesio mtrices. For exmple A d B Now A is mtrix d B is mtrix, hece the coditio for multiplyig A y B is stisfied Hece the product will e AB ) ( ) ( ) )( ( ) ( ) ( ) (

15 or AB d it is mtrix It should e oted tht if AB is defied, BA eed ot e. I the defiitio, AB would e defied oly if umer of colum of B is equl to umer or rows of A. This would llow the opertio, AB would still e differet product from BA. Becuse i the first cse A s rows re multiplied y B s colum, wheres i the secod cse B s rows re multiplied y A s colum, so tht BA d AB re completely differet. I the exmple metioed ove BA is ot possile, ecuse B is mtrix of order d A or order. The umer of colums of B is ot equl to umer or rows of A. If the product of AB of the mtrices A d B is possile d A is of order m d mtrix B of order p, the product AB is of the order m p. If A d B re oth squre mtrices of order ech the the product AB d BA re oth possile d ech of them is squre mtrix of order d they re differet. Lws of Mtrices : () Mtrix dditio is Commuttive s well s Associtive Commuttio tells the fct tht mtrix dditio clls oly for the dditio of the correspodig elemets of two mtrices d tht the order i which ech pir of correspodig elemets is dded is immteril. Commuttive lw : Commuttive lw holds good whe A + B B + A Proof : A + B ( ij ) + ( ij ) ( ij ) + ( ij ) B + A So A + B B + A Suppose A A + B B + A d B Associtive lw of dditio Associtive lw holds good whe (A+B) + C A + (B + C) Proof : (A + B) + C [( ij ) + ( ij )] + (c ij ) ( ij + ij + c ij ) ij + [( ij ) + (c ij )] A + (B + C) This c e exteded to the sum of y fiite umer of mtrices of the sme order. The sutrctio opertio A-B c simply e regrded s the dditio opertio A + (-B)

16 () I geerl Mtrix Multiplictio is ot commuttive: () Mtrix Multiplictios is Associtive d lso follows Distriutive lw. Associtive Lw of Multiplictio : (AB) C A (BC) ABC I formig the product ABC the coformility coditio must e stisfied y the mtrices. If A is m, B is p d C is p q. The A (m ) B ( p) C (p q) re coformle. Exmple : show tht the product of [x y z] h g h f g f c x y z (x + y +cz + fyz + gzx + hxy) Solutio : [x y z] h g h f g f c x y z x+ hy + gz [x y z] hx+ y + fz gx + fy + cz [x (x + hy + gz) +y (hx +y +fz) + z (gx + fy + cz)] [x + y +cz + fyz + gzx + hxy] hece provded. Distriutive Lw : Distriutive lw holds good whe A (B+C) AB + AC. ( Pre-multiplied y A) Or (B+C) A BA + CA (post-multiplied y A) I the product AB, the mtrix A is sid to e pre-multiplied y A, d the mtrix B is post multiplied y B. I BA the mtrix A is sid to e pre-multiplied y B d B to e post-multiplied y A. I ech cse, the coformility coditio for dditio s well s multiplictio must of course e oserved. Exmple : Show tht : A is ilpotet mtrix of idex.

17 M.A. (ECONOMICS) PART - I Solutio : A thus A O Hece A is ilpotet mtrix Exmple : If A show tht A where is y +ve itegr. Proof : We will prove y the method of iductio Now A if we put, we get A... Which is the sme s give vlue of A. Hece the vlue of A is true if. Where A... Now A A.A 9

18 M.A. (ECONOMICS) PART - I Hece A is true whe Assume tht vlue of A is true whe m A m m m m m A m A m m m m m m m m m m m m m m m m A m+ (Vlue otied from A o puttig m+) Hece Vlue of A is true fro m + Hece A is true wheever is y +ve iterger. Exercise :. Expli with illustrtio : (i) Cojugte Mtrix (ii) Symmetric d Skew Symmetric mtrix (iii) Orthogol Mtrix. If A, B, C so tht (ABC) C B A. Show tht the mtrix A stisfies the equtio A -A + 9A -I +

19 M.A. (ECONOMICS) PART-I (BASIC QUANTITATIVE METHODS) LESSON NO. AUTHOR : SH. C.S. AGGARWAL DETERMINANTS AND THEIR PROPERTIES There is some dete mogst ecoomists out how much oe eeds to kow out determits. It is true tht evlutig determits is rther tedious wy of solvig umericl prolems d tht theoreticl result c e otied without them. However, they re extremely useful d re widely used i the existig ecoomic literture. A determit is umer ssocited to squre mtrix. If A ( ) e mtrix the determit of A. The determit of A is writte s A or det A d is red s determit A. I other words, if A ( ij ) i,,. j,,. deotes squre mtrix of order the determit of A ij Importt Note : A determit is reducile to umer d mtrix is whole lock of umers. A determit is defied oly for squre mtrix wheres mtrix s such my ot e squre. Order of Determits. Order Oe : If A ( ) e mtrix, the det. A. Order Two : follows : If A A is mtrix of order the determit of A is defied s sclr which is otied y multiplyig the two elemets i the priciple digol of A d the sutrctig the product of the two remiig elemets. If A The A. Order Three : If A

20 is mtrix of order the determit A hs vlue : A + ( ) ( ) + ( ) + + Direct Method : There is other wy of fidig the determit of mtrix which is clled the direct method or Srrus Digrm. This method is s follows : + + This method is quite simple Write the give colums of mtrix d lso write st d d colum gi. The clculte s show ove : Multiply elemets : The sum of ll the six products will e the vlue of the determit. For Exmple : If A the A () () () + () () () + () () () () () () () () () () () ()

21 9 - For higher order determits we use Lplce s Expsio Method which is sed o co fctors. Thus we fid tht determit hs defiite vlue. The ove exmple c lso e solved y doptig the other method : A [x ()] ( ) + ( ) ( ) ( ) + ( ) (-) (-) + () Thus we fid tht the sme result is otied. Miors The determits formed y tkig equl umer of rows d colums out of the elemets of mtrix re clled miors of mtrix. The order of mior is the order of the determit. Mior is itself determit d hs vlue. I geerl, (M ij ) c e used to represet the mior otied y deletig the ith row d jth colum. Mior of First order of mtrix : A is formed y ech elemet idividully tke. Mior of Secod order of Mtrix : A is formed y tkig elemets of two rows d two colums. Mior of Third order of Mtrix : A is formed y tkig elemets of rows d colums. Cosider mtrix of order ( ) A Whe we delete y oe row d colum, which cotis the elemets of A, we get ( ) su-mtrix of A. The determit of su-mtrix is clled mior of det. A. Thus. is mior of of det. A Similrly miors of, of det. A re s follows :,, respectively For exmple, i the determit 9 the mior of elemet of i M which is M

22 Similrly, mior of elemet of M which is M 9 9 Cofctor : Aother cocept closely coected to Mior is kow s Cofctor. A Cofctor is mior with prescried sig ttched with it : The Cofctor C ij is (-) +j times the determit of the su-mtrix otied y deletig row i d colum j from mtrix A (clled the mior of ( ij ) or C ij (-) i+j M ij If the sum of the two suscripts i d j i the mior (M ij ) is eve, the the cofctor is of the sme sig s the mior : tht is (C ij ) (M ij ) if it is odd, the the cofctor tkes the opposite sig to the mior, tht is C ij (-) i+j (M ij ) (-) i+j follows the Chessord Rule i.e For exmple, for, (-) + (-) + Ad for, (-) + (-) -. Thus it is ovious tht the expressio (-) i+j c e positive oly if i + j is eve, otherwise it will e of egtive sig. It should e oted tht it is possile to expd determit y the cofctor of y row or for tht mtter, of y colum. For istce, if the first colum of third order det. A cosist of elemets,, expsio y cofctors of these elemets will lso yield tht vlue of : A [C ] + [C ] + [C ] For Exmple : Evlute the determit of the mtrix Let A The A If we expd it with the help of Ist colum

23 [C ] Cofctor of (-) + (-) - [C ] Cofctor of (-) + - (-) [C ] Cofctor of (-) + + (-) - A (-) + () + (-) A - + Similrly if we expd it with the help of first row, (-) - (-) + (-) A Properties of Determits Determits posseses certi properties tht re commo to determits of ll orders. These properties re :. The iterchge of rows d colums (i.e. trspose of mtrix), does ot effect the vlue of the determit It mes the determit of mtrix A hs the sme vlue s tht of its trspose A'. For exmple p r q s p q r s (ps qr) or -. The iterchge of two rows (or two colums) will lter the sig, ut ot the umericl vlue of the determit : For exmple p r q s (ps qr) ut with iterchge of two rows. we get r p s q (qr ps) - (ps qr) or ( ) with the iterchge of two colums, we get ( ) -. A mtrix with row (or colum) of zeros hs zero determit.

24 p q (p q) or ( ) ( ) + ( ) ( Expsio y st Row). If y sigle row (colum) of mtrix is multiplied y sclr k, the the determit is lso multiplied y k. kp r kq s (kps kqr) k (ps qr) k p r q s or Suppose k d multiplyig st colum y ( X X ) (). If oe row (or colum) is multiple of other row (or colum) the vlue of determit will e zero. For exmple p q p q pq pq or X X. If two colums (or rows) re ideticl, the determit will e zero. p p q q (pq pq) or or. The dditio (or sutrctio) of multiple of y row from other will leve the vlue of the determit ultered. For exmple p r q s (ps qr)

25 Ad p r p q s q p (s+q) (r+p) q ps + pq qr - qp (ps qr) Similrly (+) ( + ) - (studets should verify the result y sutrctio). If oe row (or colum) of mtrix is shifted to p plce (up or elow or left or right) the vlue of the determit is uchged if p is eve; the vlue of determit cquires egtive sig if p is odd : Exmple : A, chge first d secod row, the A) (-) (9-) + (9-) (-9) - (-) If oe row (or colum) of mtrix is multiplied y other row (or colum) y K the the vlue of the determit is multiplied y K: Exmple : A + Multiply st row y d rd row y : A 9 ( ) ( ) + ( 9) ( ) () + () + 9 A Thus these sic properties of determits help i simplifyig d evlutig them. Sustrctio d Iterchge of rows (or colums) further help us i reducig our work. Exercise : () Defie determit of mtrix () Defie miors d cofctors

26 . Fid the vlue of x if x. Show tht c c (-) (-c) (c-)

27 M.A. (ECONOMICS) PART-I (BASIC QUANTITATIVE METHODS) LESSON NO. AUTHOR : SH. C.S. AGGARWAL INVERSE AND RANK OF MATRICES There re two methods of fidig the iverse of give squre mtrix. (i) Guss Elimitio (or Reductio) method. (ii) Usig Adjoit mtrix/co-fctor method. i) Guss Elimitio (or Reductio) Method : If A is squre mtrix of order, I is idetity mtrix of order, the prticulr form of ew mtrix : [A/I] (y plcig I mtrix y the side of mtrix-a) is of order. A hs iverse oly d oly if [A/I] c e trsformed to [I/A - ]. The method thus cosists i plcig idetity mtrix of the sme order logside the origil mtrix A which is required to e iverted. The y performig the sme row elemetry trsformtios o oth A d I portios, we c trsform (reduce) A ito idetity mtrix, this trsformed idetify mtrix will the ecome the iverse mtrix. Exmple : Clculte iverse of mtrix : A Solutio : Plce I mtrix of order y the side of mtrix A [A/I] ~ R Applyig ~ (Applyig R R ) ~ R Applyig ~ Applyig R R

28 Which ifct is [I/A - ] A - Check if we multiply AA - we get I AA - I (i) Usig Adjoit of Mtrix : First we will defie djoit of mtrix Adjoit of A mtrix Let A ( ij ) i i i i ij i e squre mtrix of order, so tht A is the determit of the th order i.e. A.. () Let the co-fctor of A e deoted y the correspodig cpitl letters A, A etc. Thus A ij deote the cofctor of A ij i A. Let us form the mtrix of the cofctor of the correspodig cpil letter (A) d deote it y C of (A) or C (A). Thus C (A) A A A A A A A A A

29 9 C (A) my e clled cofctor mtrix. Let us tke the trspose of the mtrix C (A) so tht C (A) A A A A A A A A A ' B B B B B B B B B (sy) So tht C(A) (A ij ) (B ij ) where B ij (or A ij ) is the cofctor of ij i (A) C(A) is clled the djoit (or djugte) or mtrix A d is geerlly writte s dj. Let us ow give forml defiitio of djoit of squre mtrix. Defiitio of Adjoit of Mtrix : Let A [A ij ] e squre mtrix of order, the the trspose of the cofctor of the correspodig smll letters ij s i [A] is defied s the djoit (for djugte) of A d is riefly writte s dj. A. I simple lguge we c sy tht dj. A is the trspose of the mtrix formed y the co-fctors of the elemet of [A] or simply trspose of the cofctor mtrix. We shll illustrte the cocept with exmples : Exmple : Let A Solutio : Here [A] c c d Cofctor of + d Cofctor of - c Cofctor of c - Cofctor of d + d. Fid dj. A C (A) Cofctor Mtrix d c dj A C (A) d c

30 M.A. (ECONOMICS) PART - I Exmple : Fid djoit of A Solutio : Here A Cofctors of the elemets of the first row of A re +, -, + i.e. -, - ( 9), ( ) - Cofctors of the elemets of the secod row of (A) re -, +, i.e. - ( ), ( ) -, ( ) Cofctors of the elemets of the third row of A re +, -, + i.e. ( ), ( ) +, ( ) Now C (A) Cofctor Mtrix dj. A C (A) trspose of the cofctor mtrix Properties of djoit of A. Let A ( ij ). The product of the mtrix d its djoit is commuttive i.e. A (Adj. A) (dj. A) A A I. If A is squred sigulr mtrix (i.e. A o, the A (dj. A) (Adj. A) A [Null mtrix). dj. (AB) (dj B) (dj A) where A d B re -squred mtrices

31 . If A is symmetric, the dj A is lso symmetric.. If A is Hermiti the dj A is lso hermiti. All these properties c e verified y tkig y squre mtrix. Studets re dvised to verify these y tkig mtrix. Iverse of Mtrix (Usig Adjoit of mtrix) Let A e y squre mtrix, the mtrix B if it exists, such tht AB BA I, B is clled the iverse of A. I eig the uit mtrix d we write B A -. (A - to e red s 'A iverse ). Properties of Iverse of A. If mtrix A hs iverse, the it is uique. Let A e -squred mtrix whose iverse exists. Let us suppose tht B d C re two iverses of A, the y defiitio of iverse we hve () AB BA I B is iverse of A lso () AC CA I C is iverse of A From () d (), it follows tht AB I which implies C (AB) CI C. () Ad CA I which implies (CA) B IB B. () ut y the ssocitive lw, we kow tht C (AB) (CA) B from () d () we get B C, hece the iverse of mtrix is uique.. Coditio for squre mtrix A to possess iverse is tht A is o Sigulr i.e. A Let A e -squred mtrix d B e its iverse, the y defiitio, we hve AB I (uit mtrix) Tkig determits of oth sides AB I But AB A B d I A B I Sice R.H.S. is o zero A hs to e o zero. Hece A is o-sigulr.. If A is o-sigulr d AB AC the B C Sice A is o-sigulr A - exists Sice AB AC A - (AB) A - (AC) or (A - A) B (A - A) C or IB IC B C

32 . Reversl Lw for the iverse of the product : (AB) - A - B - i.e. the iverse of the product is the product of the iverse. The opertios of trsposig d ivertig re commuttive i.e. (A) - (A - ). (A ) - (A - ) A eig sigulr mtrix d A A cojugte trspose of A Remrks Iverse of mtrix A exists oly if (i) the give mtrix is squre mtrix d (ii) the determit of the give mtrix is o, i.e. the mtrix is o-sigulr. I other words: (i) Every mtrix eed ot hve iverse. (ii) Every squre mtrix eed t hve iverse. (iii) Every squre o-sigulr mtrix hve iverse. Let A dig (,, c) c Ad B dig,, c c The clerly AB BA Hece B is the iverse of A. Iverse of A digol mtrix : Let A dig (,, c).. Iverse of A is Digol mtrix B,, c I geerl, iverse of digol mtrix A dig (,,. ) is the digol mtrix. B dig,,...

33 How to fid Iverse of Mtrix A? We shll expli the method of fidig iverse of mtrix with the help of djoit. Let the mtrix A hs iverse B so tht A is, y defiitio o-sigulr i.e. A Now B will e iverse of A oly if it stisfies AB BA Let us choose B (dj. A) A Sice A, our choosig B s ove is justified. Now AB A A A dj. A (dj. A) A ( A I) A (A dj A A I) Similrly BA I (prove it) Hece AB BA I which shows tht B is the iverse of A. of A dj. A is the iverse of A of A - A dj. A Thus the ecessry d sufficiet coditio for mtrix A to possess iverse is tht it is o-sigulr, i.e. A # Remrks For fidig the iverse of squre mtrix A, we shll first fid the determit of A. If A, iverse does t exist. If A #, we shll fid the djoit mtrix d divide it y A to get the iverse mtrix. Exmple : Fid the iverse of A c Solutio : We hve lredy clculted dj A i exmple ove i.e. dj A d c lso A d c d

34 Assumig A to e o-zero i.e. d A - dja A d c d c c Verifictio : AA - should e Here AA - c d d c d c d c c d d c d c d c d c cd cd c d d - c -c+d Exmple : Hece A - d c d c Fid the iverse of the mtrix : Solutio : Let A - + (- - (-9) + (-) + - Cofctor of differet elemets : C (-) + + ( ) -

35 M.A. (ECONOMICS) PART - I C (-) + ( 9) + C (-) + + ( ) - C (-) + ( ) + C (-) + + ( ) - C (-) + ( ) + C (-) + + ( ) - C (-) + ( ) + C (-) + + ( ) The mtrix of co-fctor of differet elemets is Adj. A A - A A dj A

36 Exmple : Fid the djoit of the followig mtrix. Hece or otherwise fid the iverse of A Solutio : Here A A IC() C + (-) C (-) (- + ) (- + ) + (-) (- + ) () + () + (-) (-) + + we shll first fid the cofctors of the elemets of the first row of A. +, -, + i.e.,, Cofctor of the elemets of the secod row of A -, +, - i.e. - + Cofctor of the elemets of the third row of A +, -, + i.e. - Hece C (A) Cofctor of A d dj A C (A) Trspose of the cofctor mtrix

37 M.A. (ECONOMICS) PART - I Sice A [ ] A - A dj A Verifictio : AA - should e I Here AA - 9 Here A -

38 9 Exmple : Fid the iverse of A Here A C () C () + C () ( ) ( ) + ( 9) (-) (-) + (-) - + Sice A therefore A - does t exist. Remrk : Hece we eed t fid dj A. It should e oted tht we shll lwys fid A first. If A. oly the we should proceed further to fid dj. A Exmple : Fid the iverse of A Solutio : Here A

39 9 M.A. (ECONOMICS) PART - I C (A) d dj A C (A) A - A dj A A I this exmple A hs its ow iverse. Elemetry Trsformtio or Opertios : We shll ow discuss elemetry trsformtio o mtrix which will either chge its order or rk. If the trsformtio is pplied to rows, it is clled row trsformtio d if it is pplied to colums, it is clled colum trsformtio. There re three types of trsformtio s give elow :. Iterchgig of y two rows (or colums) Nottio : R ij stds for row trsformtio i which ith d jth rows of mtrix re iterchged. Similrly C ij stds for colum trsformtio i which ith d jth colum of mtrix re iterchged. For exmple if A Applyig R o A, we get B Applyig C o B, we get C Applyig C o A, we get D

40 . Nottio R i () stds for rows trsformtio i which elemets of the ith row of the mtrix re multiplied y. Similrly Ci () stds for colum trsformtio i which elemets of the i th colums of the mtrix re multiplied y. For exmple, if A Applyig R () o A, we get B. Additio to the elemets of y row (or colum) K times the correspodig elemets of y other row (or colum)_ where K is o-zero sclr. Nottio : R ij (K) stds for row trsformtio i which the elemets of the ith row multiplied y k d the dded to the correspodig elemets of the jth row. Similrly C ij (k) stds for colum trsformtio. For exmple, if A Applyig R () o A or R + R (), we get A Applyig C (-) o A, we get C Equivlet Mtrices : Two mtrices A d B re sid to e equivlet d writte s A ~ B if oe c e otied from the other y the pplictio of umer of elemetry (row of colum) trsformtios. If oly row trsformtios re pplied, the B is sid to e row equivlet to A d writte s ARB or A ~ B. Similrly if y colum trsformtio is pplied, the B is sid to e colum equivlet to A d is writte s BCA or B ~ A.

41 The Iverse Elemetry Trsformtio : If y elemetry trsformtio o mtrix A we get equivlet mtrix B, the the elemetry trsformtio which whe pplied o B gives the mtrix A, will e clled the iverse elemetry trsformtio.. Rij stds for iterchgig ith d jth rows of A d we get B. Now if o B we pply R ji we will get ck A. Thus iverse trsformtio of R ij is R ji. Similrly iverse C ij is C ji.. Ri () stds for multiplyig the ith row of A y to get B. Now if o B, pply R i we will get ck the mtrix A. Thus iverse trsformtio of Ri, () is R i iverse of C j () is C j., similrly Elemetry Mtrix A Mtrix otied from uit (or idetity) mtrix y sujectig it to y of the elemetry trsformtio is clled elemetry mtrix. Thus if I Applyig R (or C ) we get I E I E clled the elemetry mtrix, otied y iterchgig d d rd rows (colums) of I. Similrly I E () d I E () If the elemetry mtrix otied y multiplyig the d row (colum) of mtrix y is elemetry mtrix otied y multiplyig the rd row (colum) of mtrix y d the ddig it to the first row. These re the three types of elemetry mtrices. Properties :. Every elemetry row (colum) trsformtio of mtrix (ot uit mtrix) e ffected y pre-multiplictio (post-multiplictio) with the correspodig elemetry mtrix.

42 . Two mtrices A d B re equivlet if d oly if there exists o sigulr mtrices P Q such tht PAQ B. Every No sigulr squre mtrix c e expressed s the product of elemetry mtrix.. Iverse from Elemetry Mtrices : If A is reduced to y sequece of row trsformtio oly, the A - is equl to the product of correspodig elemetry mtrices i reverse order. Rk of Mtrix We shll first defie mior of mtrix. Let A e m mtrix i.e.e A ( ij ) m If we reti y r (r < m, ) rows d r colums of A we shll hve squre sumtrix of order r. The determit of the squre su-mtrix of order r is clled mior of A or order r. From the give mtrix A, we c form squre su-mtrices of order.,,, m if m < d order,,, if < m Hece the miors of mtrix A of order m will e of order,.., m if m < d of order,,.., if < m. Thus, if we hve mtrix A of order, the we c hve miors of order, d. We c t hve mior of order. For exmple let A 9. Miors of A order Ech elemet is mior of order.. Miors of A of order Reti y two rows/colums of A the determit of the squre su-mtrices or order, thus formed re clled miors of order of A. e.g. 9 9 etc. re clled miors of order of A.. Miors of A of order Reti y three rows d three colums of A d the determit of squre su-mtrices of order, thus formed re clled miors of A of order. e.g. 9 9

43 9 re miors of A of order. Rk of Mtrix Let A ( ij ) m e give mtrix of the type of mx. The rk of A, to e writte s (A), is defied to e r mi (m, ) [i.e. r is less th or equl to miimum of m d. if (i) Every mior of order r + of A is zero d (ii) There exists t lest oe mior of order r, which is o-zero. (iii) (iv) Remrk () I other words (A) r Every squre su-mtrix of order r + of A is sigulr, d There is t lest oe squre su-mtrix of order r which is o-sigulr. If oly (i) holds, it implies tht (A) r If oly (ii) holds, it implies tht (A) r If oth (i) d (ii) holds simulteously, we get (A) r () The rk of ull i.e. (zero) mtrix is zero. (c) The rk of o sigulr mtrix is lwys. (d) The rk of -squred o-sigulr mtrix is. (e) The rk of sigulr squre mtrix of order is less th. Workig Rule for Rk of Mtrix Clculte the miors of highest possile order of give mtrix A. If t lest oe of these is o zero the the order of mior is the rk. If ll the miors re zero, the clculte miors of the ext lower order. If t lest oe of them is o-zero, the this ext lower order will e the rk. If however, ll the miors of the ext order re zero, the clculte miors or still ext lower order d so o. Exmple : Fid the rk of the followig mtrices : (i) A (ii) B (iii) C (iv) D Solutio : (i) Sice A is, (A)

44 M.A. (ECONOMICS) PART - I Miors of order re M, M, M M, M, M Sice ll the miors of order re zero (A) Sice t lest oe mior of order (sy ) is o zero (A) (ii) B I () Sice B is o-sigulr (B) (iii) C (C) lso, ll miors of order re lso zero (C) But the mtrix C is o-zero mtrix d t lest oe mior of order is o zero. (C) (iv) Sice D is of order p (D) All the miors of D re i.e.,,, sice determit of ech of the mior is, (D) < ow cosider miors of D. There exist t lest oe o-zero mior of order of, viz - which is ot zero, Hece (D). Remrks : If ll the secod order miors of D hd lso ee zero, the p(d) would hve ee oe ecuse D ws o-zero mtrix. Exercise : () Fid the djoit, iverse d rk of the followig mtrices : (i) (ii) (iii)

45 M.A. (ECONOMICS) PART - I () Fid the rk of the followig mtrices : (i) 9 (ii) 9 (iii) 9 (iv) 9 The orml form of mtrix By mes of elemetry trsformtio every mtrix A of order m d rk (>) c e reduced to oe of the followig forms : (i) I r I r I r I r Exmple : Reduce the mtrix A to its orml form where A d hece determie its rk. Solutio : We shll idicte the trsformtios employed t every stge just elow the equivlece sig. So A ~ C ~ C () ~ C ()

46 M.A. (ECONOMICS) PART - I ~ C ~ R (-) ~ R ~ C, C ~ C (-), C (-) ~ I is the orml form of A. Rk (A). Exercise : Usig Guss Reductio Method or Elemetry Opertio, fid the iverse of the followig mtrices : () (). Fid rk of the followig mtrices (i) (ii)

47 M.A. (ECONOMICS) PART-I (BASIC QUANTITATIVE METHODS) LESSON NO. AUTHOR : DR. VIPLA CHOPRA SOLUTION OF SIMULTANEOUS EQUATIONS I. Itroductio II. Ojectives III. Methods for solvig simulteous Equtios. (i) Crmmer s Rule (ii) Mtrix Iverse Method IV. Summry V. Questios VI. Suggested Redigs Itroductio The stdrd form of lier equtios i two vriles x y y re x + y + c.. () x + y + c.. () These together re clled simulteous equtios s there will e oly oe pir of vlues stisfyig oth the equtios simulteously Now cosider system of lier equtios i ukows x, x. X. x + x +.. x x + x +.. x.. x + x +..+ x These equtios my e writte i the mtrix form s AX B Where A , X... x x.. x d B.. Now if A is o sigulr X A - B The result X A - B gives us solutio to the ove set of simulteous equtios Let us cosider simple cse of simulteous lier equtios x + x + x x + x + x x + x + x

48 These equtios c e expressed s AX B where A x, X x, B x II. Ojectives : The mi ojective of this lesso is to fid solutio of Lier Simulteous Equtios y usig two differet methods. III. Methods for solvig Simulteous Equtios Two methods of solvig simulteous equtios which re s follows: (i) Crmmer s Rule (ii) Mtrix Iverse Method. Crmmer s Rule : Let us cosider simple cse of simulteous equtios x + x + x. (i) x + x + x. (ii) x + x + x. (iii) Put these equtios i mtrix form s AX B where A x, X x d B x Let A ij e the co-fctor of ij i A Multiply (i), (ii) d (iii) y A, A, A respectively d dd x ( A + A + A ) + x ( A + A + A ) + x ( A + A + A ) A + A + A (iv) From the properties of determit s we kow tht the sum of products of elemets of row (or colum) with co-fctors of elemets of correspodig row (or colum) is A d sum of product of elemets of row (or colum) with co-fctors of elemets of other row (or colum) is zero From (iv), x A A + A + A i.e. x A i.e. x A + +

49 x A A, A 9 Similrly x A A, A A x A A A, A Note : To oti the umertor of vlues of x, x, x we replce y,, the elemets of first, secod d third colums respectively i A, which forms the deomitor i ech cse. Note : I geerl, we my sy tht if the determit of the mtrix of coefficiet of the system of followig lier equtios x + x + + x.. x + x + + x is ot zero, the uique solutio of X i is give y x i.. i i A.. i i The result is kow s Crmmer s Rule Thus Crmmer s Rule is ot pplicle if A Exmple : Solve the followig equtios i the mtrix form s AX B 9 x Where A, X, B x By Crmmer s Rule x A A A, x A

50 M.A. (ECONOMICS) PART - I Now A 9 A A x x x x Exmple : Cosider the followig systems of equtios : x x + x 9 x - x - x + x - x Put these equtios i mtrix form s AX B Where A, X x x x, B 9 A (- + ) + ( +) + ( -) - A 9 9 (- + ) + ( - ) + ( -) - A 9 () -9 (+) + (+)

51 A 9 ( + ) + ( +) + 9 ( - ) x A A, x A A, x A A x, x, x () Mtrix Iverse Method Aother method for solvig system of simulteous equtios is kow s mtrix iverse method. As the me suggests we hve to fid the iverse of the mtrix. Cosider system of three simulteous Lier equtios i three ukows x + x + x x + x + x x + x + x The give simulteous equtios c e writte i mtrix form s AX B Where A x, X x d B x X A - B (provided A ) Where A - is the iverse of the mtrix To oti A - we use the result A - dja A A A A A A A A A A A Now we will cosider some exmples usig mtrix iverse method to solve simulteous lier equtios. Exmple : Cosider two simulteous equtios with two vriles x y x y The give simulteous equtios c e writte i mtrix form s : AX B Where A x, X, B y

52 Now A - - (-) To oti A - we use the result A - dja A dj A Trspose of the mtrix of co-fctors A ij. Now A (-) + (-) - A (-) + () - A (-) + (-) A (-) + () Adj A A A A A A - X A - B Or x y Hece x, y Exmple : Let us cosider the followig equtios : x - y + z x + y z 9 x + y + z The ove system i the mtrix form c e writte s x y z 9 9 Now AX B X A - B... (i) We kow tht A - dja, where A A + +

53 ( +) + ( + ) + ( ) () + () + (-) A + 9 dj A trspose of A A A A A A A A A A A A A A A A A A Where : A (-) + + A (-) + - ( + ) - A (-) + ( - ) - A (-) + -(- ) A (-) + (-) A (-) + -( + ) - A (-) + ( ) A (-) + -(- ) A (-) + ( + ) Adj A A - dja A 9

54 From (), we get x y z x y z 9. X. X. X.9 X.9 X.9 X. X. X. X 9 Hece x, y, z IV. Summry : I the preset lesso we hve studied two differet methods of solvig simulteous lier equtios Uder Crmmer s Rule we put the simulteous equtios i mtrix form d the solve for the vriles. Crmmer s rule is ot pplicle if determit of the mtrix is equl to zero. Uder mtrix iverse method, gi the system of lier equtios re put i mtrix form d the y fidig iverse of the mtrix we fid the solutios. V. Questios :. Solve the followig equtios y Crmmer s Rule x + y + z, x + y + cz k, x + y + c z k. Solve the followig equtios usig Mtrix Method x - y + z x + y - z 9 x + y + z. Solve usig Mtrix Iverse Method x x + z x - y + z -x + y z. Solve the followig Lier Equtios y (usig Crmmer s Rule) : x + y z + x y + z x y z +. Solve (usig Crmmes's Rule) x + y + z x y + z x + y z

55 VI. Suggested Redigs :. A itroductio to Mthemticl Ecoomics : D. Bose. Qutittive Techiques for Mgemet : G.C. Shrm d Mdhu Ji. Mthemtics for studets of Ecoomics : Bhrdwj d Shrwl. Mthemtics for Studets of Ecoomics : Aggrwl d Joshi

56 M.A. (ECONOMICS) PART-I LESSON NO. (BASIC QUANTITATIVE METHODS) AUTHOR : DR. VIPLA CHOPRA APPLICATION OF SIMULTANEOUS EQUATIONS IN ECONOMICS I. Itroductio II. Applictio of Simulteous Equtios usig (i) Crmmer s Rule (ii) Mtrix Iverse Method III. Summry IV. Questios V. Suggested Redigs I. Itroductio I the previous lesso we hve studied two methods for solvig simulteous lier equtios i two/three vriles. Now we will solve ecoomic prolems usig simulteous equtios. II. Applictios of Simulteous Equtios usig :. Crmmer s Rule We hve lredy studied Crmmer s Rule i solvig simulteous equtios i previous lesso. Now we will cosider its pplictio i solvig some ecoomic prolems. Exmple : Cosider the followig tiol icome determitio model : Y C + I + G C + (Y T) T d + ty where Y (tiol icome) C (cosumptio expediture) d T (tx collectio) re edogeous vriles; I (Ivestmet) d G (Govt. Expediture) re exogeous vriles; t is icome tx rte. Solve for the edogeeous vriles, usig Crmmer s Rule : Sol. The Give equtios re : Y C o.t. I + G. (i) Y C.T -. (ii) ty - o.c - T -d. (iii) These c e writte s t y C T G d t () ( t) -+t t

57 G d I + G + d y I Applyig Crmmer s rule, we get Y [I +G + d] / ( + t) Similrly C d T d ( t) ( G) - t t d d t ( G) t Exmple : Three products A, B, C re produced fter eig processed through three deprtmets P, P d P. The followig dt re ville : I P I P I P A B C Mx. Time ville i hours Fid y mtrix method the umer of uit produced for ech product to hve full utiliztio of cpcity. Let,, c deote the umer of uits of products A, B, C respectively to hve full utiliztio of cpcity, the + + c + + c + + c Its mtrix form is c, i.e. AX B WhereA, X B c

58 Now A ( ) - ( ) +( ) - + By Crmmer s Rule : X A {(() ( -) + (-)} [ ] AX {(() + () - (-)} AX {(() -() + (-)} (9),, c. Mtrix Iverse Method : Exmple : The dily cost of opertig of hospitl C is lier fuctio of the umer of i-ptiets I, d out-ptiets P, plus fixed cost i.e. C + P + di Give the followig dt for dys, fid the vlues of, y settig up lier system of equtios d usig the mtrix iverse. Dy Cost (i Rs.) No. of Iptiets I No. of out Ptiets,9, 9,

59 9 Sustitutig the tulted vlues i C + P + d, We get the followig equtios : + + d, d, + + d, I the mtrix ottio, we write 9 dc,9,, Which is of the form AX B where A 9, X dc B,9,, Now A 9 ( ) - ( ) + ( 9) --+ Sice X A - B,9 A -, c () d, It c e esily verified tht dj A Sustitutig i (), we get dc,9,, 9 9 9

60 ,, d Exmple : A, B d C hve Rs., Rs. d Rs. respectively. They utilized the mouts to purchse three types of shres of prices x, y d z respectively. A Purchses shres of price x, of price y d of price z. B purchses shres of x, of price y d of price z, C purchses shre of price x, of price y d of price z. Fid x, y d z. Solutio : The followig set of simulteous lier equtios re costructed y writig give iformtio : x + y + z x + y + z x + y + z I the mtrix form, we write x y i.e. AXB z or x y z i.e.xa - B.. () Now A - dja A where A -9.. () d dj A Usig (), () d () we get :.. () x y z 9

61 /9 /9 9/9 9 9 x, y, z Exmple : A utomoile compy uses three types of steel S, S, S for producig three types of crs C, C d C. Steel requiremets (i tos) for ech types of cr re give elow : Steel Crs C C C S S S Determie the umer of crs of ech types which c e produced usig 9, d tos of steel of three types respectively. Sol. The ove iformtio c e put i the followig terms : c + c + c 9 c + c + c c + c + c Put i mtrix form we hve AC B C C C 9 Whre A, C Apply Crmmer s Rule C C, B C 9 9 9( ) - ( ) + ( )

62 ( ) - 9 ( ) + ( 9) ( ) - ( 9) + 9 ( ) ( ) - ( ) + ( ) (-) - (-) + (-) C C C C, C, C Mtrix Iverse Method : The ove exmple c e solved y usig mtrix iverse method. AC B C A - B T A A A A A A AdjA Now A -, A A A A A Now A A + ( ) -, A - ( ), A A - ( ), A ( ) -, A - ( ) A + ( ) -, A - ( 9) +, A - Adj A -

63 M.A. (ECONOMICS) PART - I A - T A - C A - B C C C 9 C C C C C C Exmple : Give the followig equilirium coditios i mrket p + p + p p + p p + p + p Fid equilirium prices i ech mrket usig Crmmer s rule d Guss elimitio method. Solutio : Give p + p + p p + p p + p + p The ove equtio c e rewritte s p + p + p p + p + p p + p + p Put i mtrix form : p p p

64 Where A, X Usig Crmmer s Rule p p, B p A ( ) - ( ) + ( ) + A ( ) - ( ) + ( ) A A P A A A, P, P A A A IV. Questios :. Solve the followig system of equtios : x x + x 9 x - x - x + x - x y (i) y the djoit method of clcultig iverse of mtrix. (ii) y mkig use of Crmmer s Rule. Suppose we re give dt o price (i Rs. per kg.) of Apples, Pottoes, Oios i the moth of Octoer, Nov. d Dec. s follows : Apples Pottoes Oios Oct. Nov... Dec.. The fmily c sped Rs., d. i Oct. Nov. d Dec. o these items. If the fmily requires to y the sme comitio of Apples, Pottoes d Oios i

65 ech moth, fid the qutity of Applies, Pottoes d Oios which the fmily uys ech of these moths.. The equilirium coditio for the relted mrkets is give y P P P -P + P P -P P + P Fid the equilirium price for ech mrket usig crmmer s rule.. Mtrix A hs X - rows d x+ colums; d mtrix B hs Y-rows d 9-Y colums. Both AB d BA defied, Fid X d Y.

66 M.A. (ECONOMICS) PART-I (BASIC QUANTITATIVE METHODS) LESSON NO. AUTHOR : DR. VIPLA CHOPRA ARITHMETIC PROGRESSION AND GEOMETRIC PROGRESSION I. Itroductio II. Ojectives III. Arithmetic Progressio (A.P.). Defiitio. th term of A.P.. Sum of terms of A.P.. Arithmetic me etwee two qutities. Arithmetic Mes (AMs) etwee d. IV. Geometric Progressio (G.P.). Defiitio. th Term of G.P.. Sum of terms of G.P.. Geometric me etwee two qutities. geometric Mes (GMs) etwee d. V. Summry VI. Questios VII.Suggested Redigs I. INTRODUCTION Before uderstdig rithmetic d geometric progressio it ecomes ecessry to uderstd the terms like sequece, series. A set of umers rrged ccordig to some defiite lw is clled sequece. For exmple,,,..., is sequece where reciprocls of ll turl umer hve ee writte i successio. A expressio cosistig of the terms of sequece joied y the sigs+ve d or-ve, is clled series. For exmple is series ssocited with the sequece,,,.,, or simply (). The vrious memers of the sequece re kow s the terms of the series. Geerlly, T deotes the th term of series d S deotes the sum of term of series. II. Ojectives : The ojectives of preset lesso re to (i) defie AP d GP, (ii) fid th terms of AP d GP,

67 (iii) fid sum of terms of AP d GP d (iv) fid A.M. d G.M. First we will ler out rithmetic progressio (A.P.) d the geometric progressio (G.P.) III. Arithmetic Progressio (A.P.) Defiitio A series i which terms icrese or decrese y commo differece is clled rithmeticl progressio. The followig series re i Arithmeticl Progressio (A.P.) :,,,, commo differece 9,,, -, -, commo differece - If first term is d commo differece is d, the series i A.P. is, + d, +d,... th Term of A.P. Let e the first term d d, the commo differece of rithmeticl progressio, the First term T + ( ) d Secod term T + d + ( ) d Third term T + d + ( ) d th term T + ( ) d Geerlly, the th term of A.P. is clled its geerl term. If A.P. cotis terms oly, the th term i.e. T is the lst term of the series d is deoted y l. T l + ( -) d. Exmple : Fid the th term of the series,,,,. Here, d -, th term i.e. T + ( ) d + ( ) (-) - Exmple : Which term of the series,,,,. is 9? Let 9 e the th term : T 9. (i) We kow T + ( )d (ii) From (i) d (ii) 9 + ( ) 9 9 is the st term. Sums of terms of A.P. Let e the first term, d the commo differece, l the lst term d S the sum of terms. S + (+d) + ( +d) +. + (l d) + (l d) + l. (i) Also S l + (l -d) + (l -d) +. + ( + d) + ( + d) +. (ii) Add (i) d (ii) we get

68 S ( + l) + ( + l) + terms ( + l) S ( + l). (I) We kow l + ( ) d Sustitutig this vlue i (I) S [ + ( ) d] Exmple : Sum up the series (i) to terms (ii) to terms (i) The give series is to terms Here.9, d , S [ + ( )d] S (ii) 9 ( ) 9 The give series is to terms Here -, d - -(-) st Term + ( ) d + d - + l Here S ( + l) (- + ) Exmple : How my terms of the series e tke to mke the sum? Here d - Let S. Fid We kow S [ + ( )d] S [ + ( ) (-)]

69 9 [ - + ] or [ ] + ( ) ( ). ()() ( ),,, Hece the umer of terms is or The series is The sum of terms will e It is self evidet tht the sum of the lst six terms is zero. The sum of terms of terms is the sme.. Arithmetic me etwee two qutities d : Whe three qutities re i A.P., the middle oe is sid to e the Arithmetic Me (A.M.) etwee the other two. Let x e the A.M. the, x, re i A.P. so tht x x x. A.M.S. etwee two qutities d : Here terms re to e iserted etwee d so tht is the first term, the ( + ) term. + ( + ) d, if d is the commo differece. d ( ) Hece the required mes re : + ( ) ( ) ( ),, ( ) ( ) ( ), ( ) Exmple : Isert rithmetic mes etwee - d Let A, A, A e A.M. s etwee - d -, A, A, A, re i A.P. -, T (totl umer of terms ) + d - + d d d

70 Now A + d A + d A + d IV. GEOMETRIC PROGRESSION (G.P.). Defiitio : A series is sid to e i Geometricl Progressio (G.P.) if the rtio of y term to the proceedig term is costt throughout. This costt fctor is clled the commo rtio. Thus :,,,.,,,,.,,,, 9, r, r,r. re ll geometric series.. th term of G.P. Let e the first term d r the commo rtio. If T deotes the th term of the series. T r - T r r - T r r - T r r T r - r - Exmple : The secod term of G.P. is the th term is, fid the series d the th term. Sol : Now secod term T d th term T As T r - T r.. (i) T r - r, r.. (ii) r Divide (ii) y (i), r, r r Put r i (i),.x, The series is,.,...

71 ,,.. T - r Exmple : if pth term of G.P. is P d qth term is Q, show tht th term of G.P. is P Q q p pq Sol. : T p r p- p.. () T q r q- Q.. () p r Dividig () y (), q r Q P, r p-q Q P, r P Q p q Put the vlue of r i (), P Q T r - put vlue of d r p q p Q T P. P P Q p p q P. P Q p p q p p q p Q P, P P p q P Q q pq p pq P Q. Sum of terms of G.P. Let e the first term, r the commo rtio The series is, r, r,.r - Let S deotes the sum of terms of the series S + r + r + + r - + r - Multiply oth sides y r. rs r + r + r + + r - + r - + r sutrct (ii) from (i) S rs r, S ( r) ( r ) S q p pq ( r ), if r < ( r). (i). (ii)

72 S (r ) (r-) ( r), if r > Note : If commo rtio r is more th oe, the we use S ( r ) th oe, the we use S ( r) Exmple : Sum up the series + +. To terms (r ) (r ) d whe r is less Here, r, S Exmple 9 : Fid, without ssumig y formul, the sum of + terms. Sol. : Here r th term x. S lso S to 9. (i). (ii) S - S - S. Geometric Me etwee two qutities Whe three qutities re i G.P., the middle oe is clled the Geometric Me (G.M.) etwee the other two.

73 M.A. (ECONOMICS) PART - I If,, c e i G.P. the c or c G.M. etwee d c is c Exmple : If,, re i G.P. the is G.M. etwee d.. G.M. s etwee two umers d Let G, G,.. G e the G.M. s etwee d, G, G,.. G re i G.P. Numer of terms + T + r +- r + r + r +, r G r G r. G r. G r. Exmple. Show tht the product of G.Ms etwee d is equl to the th power of the G.M. etwee them. Let G, G,.. G e geometric mes etwee d. G., G., G.. Now product of G.M. s G. G. G. G....

74 ..... (). ( ) G,G...G th power of G.M. etwee d Exmple : If A e the A.M. d G e the G.M. etwee two umers show tht the umers re A A G Sol. : If,, e the two umers, the A, G, +A (i) ( ) ( + ) - A G, A G i.e. Add (i) d (ii) A + A G, A A G Exmple : If,, c, d e i G.P. prove tht : ( + + c ) ( + c + d ) ( + c + cd) Solutio : Let r, c r, d r the + + c ( + r + r ) + c + d r ( + r + r ) d + c + cd r ( + r + r ) ( + r + r ). r ( + r + r ) r ( + r + r ) A G, - A G (ii)

75 Hece the result V. SUMMARY : The preset lesso defies rithmetic d geometric progressios. It lso descries the method for fidig Geerl Terms d sum of terms i cse of oth the series seprtely Besides these rithmetic mes, geometric mes hve lso ee clculted. VI. Questios : Short Questios :. Defie rithmetic progressio.. th term of A.P. is lso clled.. Whe three umers re i A.P., the middle oe is clled etwee first d lst.. Defie geometric progressio.. th term of G.P. is.. If is the first term d r is the commo rtio the sum of terms of G.P. is defied s S.. Stte the rule to fid the G.M. etwee two umers.. The product of G.M. s etwee two umrs d is the of the G.M. etwee them. Log Questios :. The sum of first terms of A.P. is. Fid the first term d the commo differece if the th term is 9.. If S, S, S e the sums of,, terms respectively of A.P., prove tht S (S S ).. Prove tht the sum of odd umer of terms of A.P. is equl to middle term multiplied y the umer of terms. VII.. The rd term of G.P. is d the th term is the reciprocl of the rd term. Fid the commo rtio. Which term of the series is uity?. Fid the vlue of so tht expressio my e the G.M. etwee d. SUGGESTED READINGS. Bhrdwj d Shrwl - Mthemtics for Studets of Ecoomics.. Aggrwl d Joshi - Mthemtics for Studets of Ecoomics.

76 M.A. (ECONOMICS) PART-I (BASIC QUANTITATIVE METHODS) LESSON NO. AUTHOR : DR. VIPLA CHOPRA ECONOMIC APPLICATION OF ARITHMETIC PROGRESSION AND GEOMETRIC PROGRESSION I. Itroductio II. Ojectives III. Applictio of Arithmetic Progressio (A.P.) IV. Applictio of Geometric Progressio (G.P.) V. Summry VI. Questios VII. Suggested Redigs I. INTRODUCTION Arithmetic d geometric progressios re frequetly used i ecoomic lysis. Ecoomic vriles like ssets, popultio etc. my chge over time either i rithmetic progressio or i geometric progressio. For exmple output of firm my icrese or decrese i rithmetic progressio over successive time periods. Geometric progressio occupies promiet plce i multiplier lysis. It is ofte ssumed tht popultio icreses i geometric progressio. Geometric me is preferred to other verges, i the mesuremet of reltive chges such s chges i the price level. Thus, rithmetic d geometric progressio hve wide pplictio i ecoomics. II. OBJECTIVES I the previous lesso we hve studied rithmetic progressio d geometric progressio i simple wy. I the preset lesso our ojective is to study ecoomic pplictio of A.P. d G.P. with illustrtios Ecoomic pplictio of (i) A.P. d (ii) G.P. III. ECONOMIC APPLICATION OF ARITHMETIC PROGRESSION Exmple : A firm produces T.V. durig its first yer. The sum totl of the firm s productio t the ed of yers ws. (i) Estimte y how my uits, productio icresed ech yer d. (ii) Forecst, sed o the estimte of the ul icremet i productio, the level of output for the th yer? Sol. : Let deote first yer s productio d d the differece etwee uits produced i two successive yers., S, (Give)

77 S [ + ( ) d] S [ + d] ( + d) + d d d Therefore productio icresed ech yer y uits. Level of output for th yer T + ( ) d Exmple : A piece of equipmet costs i certi fctory Rs.,,. If it deprecites i vlue, % the first yer, % the ext yer, % the third yer, d so o, wht will e its vlue t the ed of yers, ll percetges pplyig to the origil cost? Solutio : Let the cost of equipmet e Rs. Now the percetges of deprecitio t the ed of st, d, rd yer re,, Which re i A.P. with, d d Percetge of deprecitio i the teth yer + ( ) d + 9 Totl vlue of deprecitio i yers Hece the vlue of equipmet t the ed of yers - The totl cost eig Rs.,, its vlue t the ed of yers Rs.,, Rs.,, Exmple : A m sves Rs., i te yers. I ech yer fter the first yer he sved Rs. more th he did i proceedig yer. How much did he sved i the first yer? Here we hve to fid svigs i the first yer? umers of yer, d, S,

78 Now S [ + ( ) d] Or, [ + ( ) ], ( + 9),,, Rs., Exmple : cois re plced i stright lie o the groud. The distce etwee y two cosecutive cois is meters. How fr must perso should trvel to rig them oe y oe to sket plced meters ehid the first coi? Suppose,,, represet the positio of the cois d tht of the sket. The distce covered i rigig the First coi + The distce covered i rigig the Secod coi + The distce covered i rigig the Third coi + d so o. This form A.P. with d d Totl distce covered i rigig cois oe y oe. [ + ( ) d] [ + ( ) ] ( + ), meters. IV. ECONOMIC APPLICATION OF GEOMETRIC PROGRESSION Exmple : At % per um compoud iterest, sum of moey ccumultes to Rs. i yers. Fid the sum ivested iitilly. Let P e the priciple. Now mout of P fter yers P P 9. (.). Which is the required pricipl. P

79 9 Exmple : Give tht the mrgil propesity to cosume is, wht will e the resultt icrese i icome for utoomous itil ivestmet of Rs. lkhs? Wht will e the size of multiplier? Solutio : The mout ivested Rs. lkhs. Icrese i icome i first roud Rs. lkhs. Mrgil propesity to cosume Cosumptio i first roud Rs. As oe m s expediture is other m s icome. Icrese i icome i secod roud Rs. Similrly icrese i icome i third roud is Rs. Totl icrese i icome : d so o Rs.... Lkhs Rs.... Lkhs Rs. Multiplier Rs. Lkhs Icrese i icome Icrese i ivestmet lkhs lkhs Size of multiplier is MULTIPLIER AND GEOMETRIC SERIES I multiplier lysis, geometric series re frequetly used y ecoomists. With the help of geometric series the workig of dymic ivestmet multiplier c e esily uderstdle. Let there e repeted ivestmet of Rs. i ech of the time periods t, t.. t, Let us further ssume tht mrgil propesity to cosume is d the cosumptio i y time period depeds upo the icome i the previous

80 period. O these ssumptios icrese i icome i the vrious time periods c e illustrtes s. Period Chge i Iduced Chge Totl icrese i Ivestmet i cosumptio icome (i Rs.) t t t t I other words Icrese i icome i period t Rs. Icrese i icome i period t Rs. + Icrese i icome i period t Rs. d so o Icrese i icome i period t Rs... Rs... Rs. Rs.

81 As teds to ifiity icome teds to Rs.. I other words, s, teds to zero d therefore resultt icrese i Permet Icrese i icome Rs. Geometric Series d Popultio Growth Let P (i,,. ) e the popultio of Idi i the ith yer d let r e its rte of growth per yer. Popultio i the yer t P Popultio i the yer t P + rp, i.e P P ( + r) Similrly P P + rp P ( + r) P ( + r) P P + rp P ( + r) P ( + r) So i the yer t, t, t, respectively, the popultio P, P, P,. Form G.P. whose first term is P d commo rtio is r P i.e. popultio i the th yer P o ( + r) Exmple : The popultio of Idi ws crores i 9. Predict the popultio i 9 ssumig icrese of percet ech yer. Solutio : Popultio i yer 9 crores Icrese i popultio i ech yer % If P e the popultio i the yer t, r e the rte of growth per um, popultio i the th yer P ( + r) Popultio i the yer 9, P P ( + r) P ( +.) (.) Tke log. log P log (.) [usig log m log m + log d log m log m] log P log + log (.). + (.). +.. P Atilog (.) P. ( tilog (.).) Popultio i 9 is. crores. Exmple : A perso deposits Rs. o st Jury 99 d Rs. ech yer therefter util st Jury 99. Iterest is t the rte of % compouded ully. How much will he hve i his ccout immeditely fter he mkes the pymet o st Jury 99? Solutio : Here 9, R, i. Amout i his ccout

82 S R ( i) i 9 (.)... Exmple 9 : A od with fce vlue of Rs. mtures i yers. The omil rte of iterest o od is % p.., pid ully. Wht should e the price of the od so s to yield effective rte of retur equl to %? Solutio : We ote tht the give od yields icome strem of Rs. per yer for yers. The preset vlue of icome strem t % p.. rte of iterest is P. (.)... (.) (.) (.) (.). (.) +. (.). (.) + (.) Rs.. V. Exmple : A mchie deprecites t the rte of percet o ook vlue. The origil cost ws Rs., d ultimte scrp vlue Rs., fid the effective life of the mchie. Solutio : Origil cost of mchie, Sice mchie deprecites t the rte of % o reducig lce, 9 Vlue of mchie fter oe yer, 9 Vlue of the mchie fter two yers, Thus,, 9 9,,. will form G.P. series with 9 d r 9 Let vlue of the mchie fter yers e Rs.. T r -, 9 9

83 9 9 9 Hece the effective life of the mchie is. V. SUMMARY I the preset lesso we hve studied vrious exmples which del with the determitio of the growth of give sum of moey s well s the preset vlue of sum due t some future dte. Ofte these exmples ivolve the sum of either rithmeticl or geometricl progressio. VI. QUESTIONS. A perso sves ech yer Rs. more th the previous yer, fter te yers his svigs mout to Rs., excludig the iterest. Fid the mout the sved i the first lst yer.. A price idex ws 9 i 9 d i 9. Assumig A.P., predict the price ideed for 99. Also fid the se yer.. A firm produces uits of commodity durig the first yer of its estlishmet d the icreses the productio y uits ech yer. Fid the umer of uits produced i the th yer of the firm s history. Wht wll e the totl productio the?. Idi s popultio i 9 d 9 ws d. crore persos respectively. Fid the ul rithmetic d geometric rtes of growth.. The origil cost of mchie is Rs.,. If it deprecites t the compoud rte of % every yer, fter how my yers will it e vlued t Rs.?. The popultio of city i 9 is. If it icreses t the rte of % per um, wht will it e i 9?. If the vlue of Fit Cr deprecited y per cet ully, wht will e its estimted vlue t the ed of yers, if its preset vlue is Rs.? VII. SUGGESTED READINGS. C.S. Aggrwl d R.C. Joshi - Mthemtics for studets of Ecoomics.. G.C. Shrm d Mdhu Ji - Qutittive Techiques for Mgemet.. O.P. Bhrdwj d J.R. Shrwl - Mthemtics for studets of Ecoomics.

84 M.A. (ECONOMICS) PART-I LESSON NO. (BASIC QUANTITATIVE METHODS) AUTHOR : DR. VIPLA CHOPRA LINEAR PROGRAMMING (GRAPHICAL METHOD) Structure I. Itroductio II. Ojectives III. Lier Progrmmig (L.P.). Meig. Costituets. Assumptios. Prolem Formultio. Methods of L.P.. Grphicl Method IV. Summry V. Questios VI. Suggested Redigs I. Itroductio The cocept of Lier Progrmmig is sed upo the very importt fct tht wts or eds re ulimited while mes to stisfy them re limited. The cetrl prolem of ecoomy is how to llocte scrce/limited resources mog differet uses to get mximum possile stisfctio. Severl ecoomists took to Mrgil Alysis to solve this prolem ut Mrgil Alysis hs got its ow limittios. The method hs ot succeeded i solvig the prcticl prolems fced y ecoomy. I order to solve prcticl prolems especilly i decisio mkig y the usiess firm, the lier progrmmig techique hs ee developed. Lier Progrmmig ws first formulted y Russi mthemtici L.V. Ktorovich. But George B. Dtzig iveted superior techique of computtio i 9. It ws he who provided mthemticl frmework d computtiol method, the simplex logrithm, for formultig lier progrmmig prolems d determiig their solutios efficietly. The developmet of electroic computer hs cotriuted lot to the growth of lier progrmmig ecuse computers c quickly solve complex prolems. Lier Progrmmig is very specil su-clss of mthemticl progrmmig. It dels with the optimum use of limited resources to meet the desired ojectives d is thus helpful i rtiol decisio mkig. It is techique for solvig optimiztio (mximistio or miimistio) prolems suject to certi costrits. Thus it helps to decide whe profits shll e mximum or cost is miimum.

85 II. Ojectives : The mi ojectives of the preset lesso re to study the meig, chrcteristics d ssumptios of Lier Progrmmig. Amog differet methods of solvig lier progrmmig prolem like grphicl method, simplex method, the preset lesso dels oly with grphicl method of solvig lier progrmmig prolem. III. LINEAR PROGRAMMING. Meig Lier Progrmmig is techique for solvig mximistio d miimistios prolems suject to certi costrits. It is very specil suclss of mthemticl progrmmig. It is mthemticl techique for selectig the Best. Thus, it helps to decide whe profits shll e mximum or cost is miimum. So, L.P. is device which is used i decisio-mkig for otiig optimum vlues of qutities suject to certi costrits whe the reltioships ivolved i the prolem re lier. The term Lier implies tht ll reltioship ivolved i the prticulr prolem re lier or stright lie. The term Progrmmig refers to the process of determiig prticulr progrmme or pl of ctio. Mthemticlly, the prolem of LP my e stted s oe of optimisig Lier ojective fuctio of the followig form. Z C X + C X + C X + C i X i C X Sujectig to lier costrits of the form X + X + X + i X i X X + X + X + i X i X X + X + X + i X i X j X + x j X + j X + ji X i j X j mi X + m X + m X + mi X i m X m d X X These re clled o-egtive X i costrits X The ove LP prolem my e restted i the followig form :

86 Optimize (i.e. either mximize or miimize) Z C X i i Suject to lier costrits m i ji X i j (j,,. i m) d o-egtivity costrits X i, (i, ) where ji Ci d i ll re costts.. Costituets Every LP prolem hs some chrcteristics. The three importt chrcteristics of mthemticl form re s follows : (i) Ojective fuctios (ii) Structurl costrits (iii) No-egtive costrits (i) Ojective fuctios : There must e well defied ojective. The ojective my e mximiztio of profits or miimiztio of costs. It should e expressed s lier fuctio of decisio vrile. (ii) Structurl Costrits : I LP prolem, the costrits limit the stge to which we c pursue our ojective. These costrits must e cple of eig expressed s lier equlities or iequlities i terms of vriles. (iii) No-egtivity Costrits : They ssume tht there cot e egtive vlues of the vriles ivolved i the prolem. The o-egtivity coditios shows tht LP dels with rel life situtios for which egtive qutities re geerlly illogicl. Ojective Fuctio optimize Z C X i i j Suject to structurl Costrits m j ji X i j (j,,.m) No Negtivity Coditios X i, i,. where ji, c i d j ll re costts Exmple : Mximize z x + y. () Suject to x + y () x + y... () d x, y... () Equtio () is ojective fuctio, which tells tht if qutities A d B rig reveue Rs. d Rs. per uit, the wht mout of these qutities e j

87 produced so tht firm otis mximum reveue. Equtios () d () re structurl costrits. Equtio () is the o-egtive costrit.. Assumptios : The techique of LP is sed o some sic ssumptios. (i) Lierity : The LP prolems re lwys expressed i Lier reltio. It mes tht ojective fuctios s well s structurl costrits re lier, lso the productio reltio is lier. (ii) Proportiolity : The ssumptio of lierity gives rise to the ssumptio of proportiolity or costt returs to scle. (iii) Divisiility / Cotiuity : It mes tht i the fil solutio, s well s t other stges, we c use y qutity of iputs d produce y qutity of vrious iputs d the qutities ot ecessrily eig complete uits. (iv) Additivity : This ssumptio esures tht to fid the totl qutity of iput or output, we c hve the requiremet of ll idividul outputs. Similrly, we c fid the totl output y ddig idividul output. (v) Presece of Costrits : There must e restrictios o the mout or extet of ttimet of the ojective d these restrictio must e cple of eig expressed s lier equlities or iequlities i terms of vriles. (vi) Costt Prices : I LP opertios it is ssumed tht prices of iputs d outputs re d remi costt. It mes tht LP techique is sed o purely competitive pproch to prices. (vii) Fiiteess : It is ssumed tht the umer of ctivities d resources will e limited. If we cosider idefitely lrge umer of ctivities d resources, there will ot e y ed of computtiol work d fidig out optimum solutio will e rther impossile. (viii) Sigle vlue Expecttios : The resource supplies, iputs output coefficiets, d prices re kow with certity. I frmig situtios sometimes this ssumptio is foud to e urelistic e.g. we cot sy with certity s wht will e the yield of crop i plig period or wht prices will previl i the mrket for vrious resources d outputs. But still the use of LP remis meigful s severl techiques re ville to predict the future evets quite precisely d miimizig the ucertity.. Prolem Formultio i Lier Progrmmig Let us cosider profit mximisig firm which possesses fixed qutities of cpcity i m-opertios. The firm produces X, X, X, X.. X m commodities which rig P, P, P, P.. P of reveue per uit of commodity X, X, X, X respectively.

88 Opertio (Dept) Time Tke y Products Mx. Cpcity X X X X X ville st uits d. uits rd. uits th mth m m m m. m uits Reveue P P P P...P per uit Product Now the totl et reveue of the firm which eeds to e mximised will e R X P + X P + X P +.. +X P Now the totl cpcity of opertio I cosumed y uits X (i) (ii) Similrly The totl cpcity of opertio cosumed y X uits X The totl cpcity ville i Opertio I is t most uits i.e. X + X + X X must ot e more th X + X.. + X Smewy X + X.. + X m X + m X.. + m X Let us ow put the prolem i its stdrd form : Ojective fuctio Mximize P X + P X.. + P X Suject to Costrits X + X.. + X X + X.. + X m X + m X.. + m X m No egtive Costrits X, X.. X Miimiztio Progrmme Mi z P X + P X..+ P X.. ojective fuctio suject to certi costrits X + X.. + X X + X.. + X

89 9 costrits m X + m X.. + m X m X, X.. X No egtive costrits Writig Lier Progrmmig By Nottio method Optimize (i.e. either mximize or miimize) z p X i i Suject to lier Costrits p X i i (j,,,.. m) i j j d o-egtivity costrits x i o, (i,,.. ) where ji, c i d j ll the costts. Miimiztio z p X i i Suject to : ij x j i j j i (i,,.. m) x j. Methods of Lier Progrmmig A Lier Progrmmig prolem c e solved either y the grphicl method or y simplex method. Grphicl method is quite simple d is geerlly used whe there re oly two choice vriles i the ojective fuctio. Now we shll discuss grphicl method of solvig L.P. prolem.. Grphicl Method If the ojective fuctio z is fuctio of two vriles oly the the prolem c e solved y grphicl method. Grphicl method of LP ivolves three steps plottig the differet costrits o, the grph pper, fidig the fesiility regio d its extreme poits d filly testig which extreme poit yields optimum solutio. To illustrte the use of the grphicl method to solve L.P. prolem, we shll discuss few exmples. Exmple : Mx. Z X + x Suject to : x + x x + x d x, x Sol. First step is to plot lier costrits o the grph pper, Product I is plotted log the x-xis d product II is plotted log the y-xis. To plot first costrit x + x o the grph pper we require two poits. Suppose x i.e. The etire mout of iput A is used to produce the product II. Whe X the x

90 9. I this wy we get the first poit (, ). This poit P o grph deotes zero productio of product d uits productio of product II. To fid the secod poit let us ssume tht the etire mout of iput A is used to produce the product I d o uit of product II is produced, i.e. x. Therefore x i.e. the firm c produce either or less th uits of product I. If we tke the mximum productio the x. So the secod poit is (, ) d this poit Q i grph deotes productio of uits of product I d zero uits of product II. By joiig P d Q we get stright lie PQ. This lie shows the mximum qutities of product I d product II. The re POQ is the grphic represettio of costrit x + x I similr wy the secod costrit x + x c e drw grphiclly. For this purpose we oti two poits R (, ) d S (, ). The lie RS i Fig. represets the mximum qutities of product I d II tht c e produced

91 9 with the help of iput B. ROS represets the fesiility regio s fr s the iput B is cocered. After plotig the two costrits ext step is to fid out the fesiility regio. (Fesiility) regio is tht regio i the grph which sttisfies ll the costrits simulteously. Regio POST i third grph which sttisfies ll the costrits simulteously. ROS is the fesile regio uder first costrit. But out of this RPT regio does ot fulfill secod costrit. I the sme wy regio POQ is the fesiility uder costrit secod ut out of this, regio TSQ does ot stisfy costrit first. Regio POST i Fig. thus, stisfies first costrit s well s secod costrit d is thus fesile regio. Ech poit i the fesile regio POST stisfies oth the lier costrits d is therefore fesile solutio. No-egtive costrits re lso eig stisfied i this regio ecuse we re tkig qudrdt of the grph i which oth the xes re positive. The fesile regio is covered y the kiked oudry PTS. The corer poits o the kiked oudry P, T d S re clled s extreme poits. Let us see which of these three poits, P, T d S yield mximum profit to the firm. At Poit p, x, x Z x + x () + () Rs. At Poit T, x, x Z x + x () + () Rs. At Poit S, x, x Z x + x () + () Rs. So comitio T (, ) is the optimum d Rs. is the mximum possile level of profits.

92 9 Exmple : Mximize Z x + x Suject to the costrits x + x x + x x x, x Step. Costruct the grph, Cosider set of rectgulr Crtesi xis OX, X i the ple. Ech poit hs coordites of the type (x x ) d coversely every ordered pir (x x ) of rel umers determies poit i the ple. It is cler tht y poit which stisfies the coditio x d x lies i the first qudrt d coversely for y poit (x,x ) i the first qurdrt x d x Step. To grph ech costrit i the first qurdt stisfyig the costrits, we first tret ech iequtio s though it were equtio d the fid the set of poits i the first qudrt stisfyig the costrits. Ay poits stisfyig the iequtio x + x lies i the first qurdrt o tht of the lie x + x which cotits the origi. I similr wy, we see tht ll poits stisfyig the costrit x + x re the poits i the first qudrt lyig o or elow the lie x + x. The set of poits stisfyig the iequtio x lies o or towrds the left of the lie x. Step. All poits i the re show shded i Fig. stisfy the three costrits d lso the o-egtive restrictios x, x. X D X X + X B X + X B O A 9 X Fig. No.

93 9 This shded re is clled the covex regio or the solutio spce or the regio of fesile solutios. Ay poit i this regio is fesile solutio to the give prolem. The covex OABCD is ouded y the lies x, x, x + x, x + x d x. The five vertices of the covex regio re O (,), A (, ), B (, ), C (, ) d D (, ) The solutio to the prolem is x, x d mximum z. Exmple : Miimize C x + x Suject to x + x x + x x + x d x, x Step I Represet the ove iequlities i the grph pper. Cosider the first costrit. x + x Fid few poits. By puttig x we get x Similrly y puttig x, we get x. Hece we otied two poits (, ) d (, ). By joiig these two poits, we oti stright lie represetig the ove equtio. Similrly other costrits re show i the grph i Fig..

94 9 Step II. Mrk the fesile regio. Here fesile regio cosists of y poit o or ove thick lie ABCD. Step III. Locte the corer solutio. These pper t A, B, C d D. IV. Summry I this lesso we hve studied the techique of LP for solvig mximistio d miimistio prolems suject to certi costrits. We hve lso discussed the mi chrcteristics d ssumptios of L.P. prolem. Grphicl method for solvig L.P. prolem hs ee discussed i detil with illustrtios. V. Questios Short Questios (i) Defie Lier Progrmmig. (ii) Stte two methods of solvig Lier Progrmmig prolem. (iii) Stte two ssumptios of L.P. (iv) Wht do you me y structurl costrits? (v) Metio three importt chrcteristics of Lier Progrmmig. Log Questios. Miimize totl cost.x +. x Suject to x + x x + x x + x d x, x. Mx. z x + x, suject to costrits. x, x x + x x + x, x, x. Mximize Z x + x, suject to costrits x, x x x, x, x. A firm mkes two types of furiture : chirs d tles. The cotriutio to profit for ech product s clculted y the ccoutig deprtmet is Rs. per chir d Rs. per tle. Both product re to e processed o three mchies M, M d M. The time required i hours per week o ech mchie re s follows : Mchie Chir Tle Aville Time M M M

95 9 How should the mufctures schedule his productio i order to mximise profit? (i) Give mthemticl formultio to this Lier Progrmmig Prolem. (ii) Use grphicl method to solve this prolem. VI. Suggested Redigs :. Aggrwl d Joshi : Mthemtics for studets of Ecoomics.. G. Hdeley : Lier Progrmmig. Sul I. Gss : Lier Progrmmig methods d pplictios.. Meht-Mdi : Mthemtics for Ecoomists. Type Settig : Computer L, Dept, of Distce Eductio, Puji Uiversity, Ptil.

96 Puji Uiversity, Ptil M.A. (ECONOMICS) PART-I (Semester-I) PAPER-III BASIC QUANTITATIVE METHODS ACADEMIC SESSION : - Sectio - B SECTION - B Lesso No : Deprtmet of Distce Eductio 9. Mtrices. Determits d their properties. Iverse d Rk of Mtrices (All Copyrights re Reserved). Solutio of Simulteous Equtios. Applictio of Simulteous Equtios i Ecoomics. Arithmetic Progressio d Geometric Progressio. Ecoomic Applictio of Arithmtic Progressio d Geometric Progressio. Lier Progrmmig (Grphicl Method)