BUSINESS MATHEMATICS

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1 BUSINESS MATHEMATICS HIGHER SECONDARY - SECOND YEAR olume- Untouchbility is sin Untouchbility is crime Untouchbility is inhumn TAMILNADU TEXTBOOK CORORATION College Rod, Chenni

2 Government of Tmilndu First Edition - Second Edition - 6 Tet Book Committee Chirperson Dr.S.ANTONY RAJ Reder in Mthemtics residency College Chenni 6. Thiru. N. RAMESH Selection Grde Lecturer Deprtment of Mthemtics Govt. Arts College (Men) Nndnm, Chenni - 6. Thiru.. RAKASH Lecturer (S.S.), Deprtment of Sttistics residency College Chenni - 6. Thiru. S.T. ADMANABHAN ost Grdute Techer The Hindu Hr. Sec. School Triplicne, Chenni - 6. Tmt. AMALI RAJA ost Grdute Techer Good Shepherd Mtricultion Hr. Sec. School, Chenni 66. rice : Rs. Reviewers - cum - Authors Authors Reviewer Dr. M.R. SRINIASAN Reder in Sttistics University of Mdrs, Chenni - 6. Thiru. R.MURTHY Selection Grde Lecturer Deprtment of Mthemtics residency College Chenni 6. Thiru. S. RAMACHANDRAN ost Grdute Techer The Chintdripet Hr. Sec. School Chintdripet, Chenni - 6. Thiru. S. RAMAN ost Grdute Techer Jigopl Grodi Ntionl Hr. Sec. School Est Tmbrm, Chenni Tmt. M.MALINI ost Grdute Techer.S. Hr. Sec. School (Min) Mylpore, Chenni 6. This book hs been prepred by the Directorte of School Eduction on behlf of the Government of Tmilndu This book hs been printed on 6 GSM pper

3 refce The most distinct nd beutiful sttement of ny truth must tlst tke the Mthemticl form -Thoreu. Among the Nobel Luretes in Economics more thn 6% were Economists who hve done pioneering work in Mthemticl Economics.These Economists not only lernt Higher Mthemtics with perfection but lso pplied it successfully in their higher pursuits of both Mcroeconomics nd Econometrics. A Mthemticl formul (involving stochstic differentil equtions) ws discovered in 97 by Stnford University rofessor of Finnce Dr.Scholes nd Economist Dr.Merton.This chievement led to their winning Nobel rize for Economics in 997.This formul tkes four input vribles-durtion of the option,prices,interest rtes nd mrket voltility-nd produces price tht should be chrged for the option.not only did the formul work,it trnsformed Americn Stock Mrket. Economics ws considered s deductive science using verbl logic grounded on few bsic ioms.but tody the trnsformtion of Economics is complete.etensive use of grphs,equtions nd Sttistics replced the verbl deductive method.mthemtics is used in Economics by beginning wth few vribles,grdully introducing other vribles nd then deriving the inter reltions nd the internl logic of n economic model.thus Economic knowledge cn be discovered nd etended by mens of mthemticl formultions. Modern Risk Mngement including Insurnce,Stock Trding nd Investment depend on Mthemtics nd it is fct tht one cn use Mthemtics dvntgeously to predict the future with more precision!not with % ccurcy, of course.but well enough so tht one cn mke wise decision s to where to invest money.the ide of using Mthemtics to predict the future goes bck to two 7 th Century French Mthemticins scl nd Fermt.They worked out probbilities of the vrious outcomes in gme where two dice re thrown fied number of times. iii

4 In view of the incresing compleity of modern economic problems,the need to lern nd eplore the possibilities of the new methods is becoming ever more pressing.if methods bsed on Mthemtics nd Sttistics re used suitbly ccording to the needs of Socil Sciences they cn prove to be compct, consistent nd powerful tools especilly in the fields of Economics, Commerce nd Industry. Further these methods not only gurntee deeper insight into the subject but lso led us towrds ect nd nlyticl solutions to problems treted. This tet book hs been designed in conformity with the revised syllbus of Business Mthemtics(XII) (to come into force from - 6)- Ech topic is developed systemticlly rigorously treted from first principles nd mny worked out emples re provided t every stge to enble the students grsp the concepts nd terminology nd equip themselves to encounter problems. Questions compiled in the Eercises will provide students sufficient prctice nd self confidence. Students re dvised to red nd simultneously dopt pen nd pper for crrying out ctul mthemticl clcultions step by step. As the Sttistics component of this Tet Book involves problems bsed on numericl clcultions,business Mthemtics students re dvised to use clcultors.those students who succeed in solving the problems on their own efforts will surely find phenomenl increse in their knowledge, understnding cpcity nd problem solving bility. They will find it effortless to reproduce the solutions in the ublic Emintion. We thnk the Almighty God for blessing our endevour nd we do hope tht the cdemic community will find this tetbook triggering their interests on the subject! The direct ppliction of Mthemticl resoning to the discovery of economic truth hs recently rendered gret services in the hnds of mster Mthemticins Alfred Mrshll. Mlini Amli Rj Rmn dmnbhn Rmchndrn rksh Murthy Rmesh Srinivsn Antony Rj iv

5 CONTENTS ge. ALICATIONS OF MATRICES AND DETERMINANTS. Inverse of Mtri Minors nd Cofctors of the elements of determinnt - Adjoint of squre mtri - Inverse of non singulr mtri.. Systems of liner equtions Submtrices nd minors of mtri - Rnk of mtri - Elementry opertions nd equivlent mtrices - Systems of liner equtions - Consistency of equtions - Testing the consistency of equtions by rnk method.. Solution of liner equtions Solution by Mtri method - Solution by Crmer s rule. Storing Informtion Reltion mtrices - Route Mtrices - Cryptogrphy. Input - Output Anlysis.6 Trnsition robbility Mtrices. ANALYTICAL GEOMETRY 66. Conics The generl eqution of conic. rbol Stndrd eqution of prbol - Trcing of the prbol. Ellipse Stndrd eqution of ellipse - Trcing of the ellipse. Hyperbol Stndrd eqution of hyperbol - Trcing of the hyperbol - Asymptotes - Rectngulr hyperbol - Stndrd eqution of rectngulr hyperbol. ALICATIONS OF DIFFERENTIATION - I 99. Functions in economics nd commerce Demnd function - Supply function - Cost function - Revenue function - rofit function - Elsticity - Elsticity of demnd - Elsticity of supply - Equilibrium price - Equilibrium quntity - Reltion between mrginl revenue nd elsticity of demnd. v

6 . Derivtive s rte of chnge Rte of chnge of quntity - Relted rtes of chnge. Derivtive s mesure of slope Slope of the tngent line - Eqution of the tngent - Eqution of the norml. ALICATIONS OF DIFFERENTIATION - II. Mim nd Minim Incresing nd decresing functions - Sign of the derivtive - Sttionry vlue of function - Mimum nd minimum vlues - Locl nd globl mim nd minim - Criteri for mim nd minim - Concvity nd conveity - Conditions for concvity nd conveity - oint of inflection - Conditions for point of inflection.. Appliction of Mim nd Minim Inventory control - Costs involved in inventory problems - Economic order quntity - Wilson s economic order quntity formul.. rtil Derivtives Definition - Successive prtil derivtives - Homogeneous functions - Euler s theorem on Homogeneous functions.. Applictions of rtil Derivtives roduction function - Mrginl productivities - rtil Elsticities of demnd.. ALICATIONS OF INTEGRATION 7. Fundmentl Theorem of Integrl Clculus roperties of definite integrls. Geometricl Interprettion of Definite Integrl s Are Under Curve. Appliction of Integrtion in Economics nd Commerce The cost function nd verge cost function from mrginl cost function - The revenue function nd demnd function from mrginl revenue function - The demnd function from elsticity of demnd.. Consumers Surplus. roducers Surplus ANSWERS 7 (...continued in olume-) vi

7 ALICATIONS OF MATRICES AND DETERMINANTS The concept of mtrices nd determinnts hs etensive pplictions in mny fields such s Economics, Commerce nd Industry. In this chpter we shll develop some new techniques bsed on mtrices nd determinnts nd discuss their pplictions.. INERSE OF A MATRIX.. Minors nd Cofctors of the elements of determinnt. The minor of n element ij of determinnt A is denoted by M i j nd is the determinnt obtined from A by deleting the row nd the column where i j occurs. The cofctor of n element ij with minor M ij is denoted by C ij nd is defined s Mi j, C ij M, i j if if i + j is even i + j is odd. Thus, cofctors re signed minors. In the cse of, we hve M, M, M, M Also C, C, C, C In the cse of M M, C, we hve, C ; ;

8 M M, C, C ; nd so on... Adjoint of squre mtri. The trnspose of the mtri got by replcing ll the elements of squre mtri A by their corresponding cofctors in A is clled the Adjoint of A or Adjugte of A nd is denoted by Adj A. Thus, Note AdjA A t c b d (i) Let A then A c d c b Adj A A t c d b c Thus the Adjoint of mtri cn be written instntly s d c b c c (ii) Adj I I, where I is the unit mtri. (iii) A(AdjA) (Adj A) A A I (iv) Adj (AB) (Adj B) (Adj A) (v) If A is squre mtri of order, then AdjA A If A is squre mtri of order, then Adj A A Emple Write the Adjoint of the mtri A Adj A Emple Find the Adjoint of the mtri A b d

9 A Now,, Adj A A t c C, C 8, C, C, C 6, C, C, C, C Hence A c Adj A t Inverse of non singulr mtri. The inverse of non singulr mtri A is the mtri B such tht AB BA I. B is then clled the inverse of A nd denoted by A. Note (i) A non squre mtri hs no inverse. (ii) The inverse of squre mtri A eists only when A tht is, if A is singulr mtri then A does not eist. (iii) If B is the inverse of A then A is the inverse of B. Tht is B A A B. (iv) A A I A - A (v) The inverse of mtri, if it eists, is unique. Tht is, no mtri cn hve more thn one inverse. (vi) The order of the mtri A will be the sme s tht of A.

10 (vii) I I (viii) (AB) B A, provided the inverses eist. (i) A I implies A A () If AB C then () A CB (b) B A C, provided the inverses eist. (i) We hve seen tht A(AdjA) (AdjA)A A I A (AdjA) (AdjA)A I (ΠA ) A A This suggests tht A A (AdjA). Tht is, A A A t c (ii) instntly s (A ) A, provided the inverse eists. b Let A Now A c Emple ~ A c d b d c d bc with A d bc, A t c d c b d c Thus the inverse of mtri d bc d c b Find the inverse of A A A b c b d provided d bc. A eists., if it eists. cn be written

11 Emple Show tht the inverses of the following do not eist : (i) A 9 6 (ii) A 6 7 (i) A 9 6 A does not eist. (ii) A 6 7 A - does not eist. Emple Find the inverse of A, if it eists. A A eists. We hve, A A A t c Now, the cofctors re C, C 7, C, C, C 8, C, C, C, C, Hence A c 8 7, A t c 8 7 A 8 7

12 6 Emple 6 Show tht A nd B re inverse of ech other. AB I Since A nd B re squre mtrices nd AB I, A nd B re inverse of ech other. EXERCISE. ) Find the Adjoint of the mtri ) Find the Adjoint of the mtri ) Show tht the Adjoint of the mtri A is A itself. ) If A, verify tht A(Adj A) (Adj A) A A I. ) Given A, B, verify tht Adj (AB) (Adj B) (Adj A)

13 7 6) In the second order mtri A ( i j ), given tht i j i+j, write out the mtri A nd verify tht Adj A A 7) Given A -, verify tht Adj A A 8) Write the inverse of A 9) Find the inverse of A ) Find the inverse of A b nd verify tht AA I. ) If A nd none of the s re zero, find A. ) If A, show tht the inverse of A is itself. ) If A, find A. ) Show tht A nd B re inverse of ech other ) If A 8, compute A nd show tht A I A 6) If A verfy tht (A ) A 7) erify (AB) B A, when A nd B 9 6 8) Find λ if the mtri ë 9 ë 7 6 hs no inverse.

14 8 9) If X 6 nd Y q p find p, q such tht Y X. ) If X 9, find the mtri X.. SYSTEMS OF LINEAR EQUATIONS.. Submtrices nd minors of mtri. Mtrices obtined from given mtri A by omitting some of its rows nd columns re clled sub mtrices of A. e.g. If A, some of the submtrices of A re :,,,,,, nd The determinnts of the squre submtrices re clled minors of the mtri. Some of the minors of A re :,,,, nd.. Rnk of mtri. A positive integer r is sid to be the rnk of non zero mtri A, denoted by ρ(a), if (i) there is t lest one minor of A of order r which is not zero nd (ii) every minor of A of order greter thn r is zero.

15 Note (i) The rnk of mtri A is the order of the lrgest non zero minor of A. (ii) If A is mtri of order m n then ρ(a) < minimum (m, n) (iii) The rnk of zero mtri is tken to be. (iv) For non zero mtrices, the lest vlue of the rnk is. (v) The rnk of non singulr mtri of order n n is n. (vi) ρ(a) ρ(a t ) (vii) ρ(i ), ρ(i ) Emple 7 Find the rnk of the mtri A Order of A is. ρ(a) < Consider the only third order minor Emple 8. 9 There is minor of order which is not zero. ρ(a) Find the rnk of the mtri A Order of A is. ρ(a) < Consider the only third order minor 6 6 The only minor of order is zero. ρ(a) < Consider the second order minors.

16 We find, Emple 9 There is minor of order which is non zero. ρ(a). Find the rnk of the mtri A 8 6 Order of A is. ρ(a) < Consider the only third order minor 6 8 (R R ) The only minor of order is zero. ρ(a) < Consider the second order minors. Obviously they re ll zero. ρ(a) < Since A is non zero mtri, ρ(a) Emple Find the rnk of the mtri A 9 Order of A is. ρ(a) < Consider the second order minors. We find, 9 8 There is minor of order which is not zero. ρ(a) 7 Emple Find the rnk of the mtri A

17 Order of A is. ρ(a) <. We find, Consider the third order minors. 8 9 There is minor of order which is not zero. ρ(a)... Elementry opertions nd equivlent mtrices. The process of finding the vlues of number of minors in our endevour to find the rnk of mtri becomes lborious unless by stroke of luck we get non zero minor t n erly stge. To get over this difficulty, we introduce mny zeros in the mtri by wht re clled elementry opertions so tht the evlution of the minors is rendered esier. It cn be proved tht the elementry opertions do not lter the rnk of mtri. (i) (ii) (iii) The following re the elementry opertions : The interchnge of two rows. The multipliction of row by non zero number. The ddition of multiple of one row to nother row. If mtri B is obtined from mtri A by finite number of elementry opertions then we sy tht the mtrices A nd B re equivlent mtrices nd we write A B. Also, while introducing mny zeros in the given mtri, it would be desirble (but not necessry) to reduce it to tringulr form. A mtri A ( i j ) is sid to be in tringulr form if i j whenever i > j. e.g., The mtri 7 9 is in tringulr form.

18 Emple Find the rnk of the mtri A - Order of A is. ρ(a) <. Let us reduce the mtri A to tringulr form. A Applying R R A ~ Applying R R R A ~ 8 Applying R R 8R A ~ This is now in tringulr form. We find, There is minor of order which is not zero. ρ(a). Emple Find the rnk of the mtri A

19 Order of A is. ρ(a) < Let us reduce the mtri A to tringulr form. A Applying R R R, R R R A ~ Applying R R + R A 8 This is now in tringulr form. We find, 8 6 There is minor of order which is not zero. ρ(a). Emple Find the rnk of the mtri A 8 6 Order of A is. ρ (A) <. A 8 6 Applying R R

20 A 6 Applying R R A 6 Applying R R R, R R R A ~ 6 6 Applying R R + R A ~ 8 6 This is in tringulr form. We find, There is minor of order which is not zero. ρ(a).. Systems of liner equtions. A system of (simultneous) equtions in which the vribles (ie. the unknowns) occur only in the first degree is sid to be liner. A system of liner equtions cn be represented in the form AX B. For emple, the equtions y+z, +yz, 67y+8z 7 cn be written in the mtri form s z y 7 A X B

21 A is clled the coefficient mtri. If the mtri A is ugmented with the column mtri B, t the end, we get the ugmented mtri, M M denoted by (A, B) M 7 A system of (simultneous) liner equtions is sid to be homogeneous if the constnt term in ech of the equtions is zero. A system of liner homogeneous equtions cn be represented in the form AX O. For emple, the equtions +yz, +y, y+z cn be written in the mtri form s y z A X O.. Consistency of equtions A system of equtions is sid to be consistent if it hs t lest one set of solution. Otherwise it is sid to be inconsistent. Consistent equtions my hve (i) unique solution (tht is, only one set of solution) or (ii) infinite sets of solution. By wy of illustrtion, consider first the cse of liner equtions in two vribles. The equtions y 8, + y represent two stright lines intersecting t (, ). They re consistent nd hve the unique solution, y. (Fig..) y -y 8 (, ) +y Consistent ; Unique solution O Fig..

22 The equtions y, y represent two coincident lines. We find tht ny point on the line is solution. The equtions re consistent nd hve infinite sets of solution such s, y - ;, y ;, y nd so on (Fig..) Such equtions re clled dependent equtions. y O Consistent ; Infinite sets of solution. The equtions y, 8 y represent two prllel stright lines. The equtions re inconsistent nd hve no solution. (Fig..) y. (, ). (, -). (, ) -y, -y Fig.. O 8 - y - y Fig.. Inconsistent ; No solution Now consider the cse of liner equtions in three vribles. The equtions + y + z, + y + z 6, + y + z 6 re consistent nd hve only one set of unique solution viz., y, z. On the other hnd, the equtions + y + z, + y + z, + y + z re consistent nd hve infinite sets of solution such s, y, z ;, y -, z ; nd so on. All these solutions re included in +k, y -k, z k where k is prmeter. 6

23 The equtions + y + z, + y z -, +y + 7z 7 do not hve even single set of solution. They re inconsistent. All homogeneous equtions do hve the trivil solution, y, z. Hence the homogeneous equtions re ll consistent nd the question of their being consistent or otherwise does not rise t ll. The homogeneous equtions my or my not hve solutions other thn the trivil solution. For emple, the equtions + y + z, y z, +y z hve only the trivil solution viz.,, y, z. On the other hnd the equtions +y -z, y +z, +6y -z hve infinite sets of solution such s, y, z ;, y 6, z 9 nd so on. All these non trivil solutions re included in t, y t, z t where t is prmeter...6 Testing the consistency of equtions by rnk method. Consider the equtions AX B in 'n' unknowns ) If ρ(a, B) ρ(a), then the equtions re consistent. ) If ρ(a, B) ρ(a), then the equtions re inconsistent. ) If ρ(a, B) ρ(a) n, then the equtions re consistent nd hve unique solution. ) If ρ(a, B) ρ(a) < n, then the equtions re consistent nd hve infinite sets of solution. Consider the equtions AX in 'n' unkowns ) If ρ(a) n then equtions hve the trivil solution only. ) If ρ(a) < n then equtions hve the non trivil solutions lso. Emple Show tht the equtions -y +z 7, +y-z, +y +z re consistent nd hve unique solution. 7

24 8 The equtions tke the mtri form s z y 7 A X B Now (A, B) 7 M M M Applying R R (A, B) 7 M M M Applying R R R, R R R (A, B) 8 M M M Applying R R R (A, B) 8 M M M Obviously, ρ(a, B), ρ(a) The number of unknowns is. Hence ρ(a, B) ρ(a) the number of unknowns. The equtions re consistent nd hve unique solution. Emple 6 Show tht the equtions + y, y - z, + y + z re consistent nd hve infinite sets of solution. The equtions tke the mtri form s

25 9 - z y A X B Now, (A, B) - M M M Applying R R - R (A, B) M M M Applying R R +R (A, B) ~ - M M M Obviously, ρ(a, B), ρ(a). The number of unknowns is. Hence ρ(a, B) ρ(a) < the number of unknowns. The equtions re consistent nd hve infinite sets of solution. Emple 7 Show tht the equtions -y +z, -y +7z 6, -8y +z re inconsistent. The equtions tke the mtri form s 8 7 z y 6 A X B

26 Now, (A, B) M M M Applying R R - R, R R -R (A, B) ~ 8 M M M Applying R R -R (A, B) ~ 8 M M M Obviously, ρ(a, B), ρ(a) Hence ρ(a, B) ρ(a) The equtions re inconsistent. Empe 8 Show tht the equtions +y +z, +y -z, -y +z hve only the trivil solution. The mtri form of the equtions is z y A X O A Applying R R R, R R R

27 A Applying R R R A ~ 9 Obviously, ρ (A) The number of unknowns is. Hence ρ (A) the number of unknowns. The equtions hve only the trivil solution. Emple 9 Show tht the equtions +y +9z, +y +z, +y +7z hve non trivil solutions lso. The mtri form of the equtions is 7 9 z y A X O A 7 9 A 7 9, ρ (A) The number of unknowns is. Hence ρ(a) < the number of unknowns. The equtions hve non trivil solutions lso.

28 Emple Find k if the equtions + y -z, -y +z, +7y -6z k re consistent. (A, B) k 6 7 M M M, A 6 7 A 6 7, Obviously ρ(a). For the equtions to be consistent, ρ(a, B) should lso be. Hence every minor of (A, B) of order should be zero. k 6 7 Epnding nd simplifying, we get k 8. Emple Find k if the equtions + y + z, +y +z 6, +y +z k re inconsistent. (A, B) k M M M 6, A We find, A, Obviously ρ(a). For the equtions to be inconsistent, ρ(a, B) should not be.

29 (A, B) k M M M 6 Applying R R R, R R R (A, B) ~ k M M M Applying R R R (A, B) ~ 9 k M M M ρ(a, B) only when k 9 The equtions re inconsistent when k ssumes ny rel vlue other thn 9. Emple Find the vlue of k for the equtions k + y + z, -y + z, k - y + z to hve non trivil solution. A k k For the homogeneous equtions to hve non trivil solution, ρ(a) should be less thn the number of unknowns viz.,. ρ(a). Hence k k Epnding nd simplifying, we get k Emple Find k if the equtions + y +z, -y -z, +y +kz hve only the trivil solution.

30 A k For the homogeneous equtions to hve only the trivil solution, ρ(a) should be equl to the number of unknowns viz.,. k, k. The equtions hve only the trivil solution when k ssumes ny rel vlue other thn. EXERCISE. ) Find the rnk of ech of the following mtrices (i) (ii) 6 6 (iii) (iv) 7 (v) 8 6 (vi) 7 (vii) 9 9 (viii) (i) ) Find the rnks of A+B nd AB where A nd B 6 6 ) rove tht the points (, y ), (, y ) nd (, y ) re colliner if the rnk of the mtri y y y is less thn. ) Show tht the equtions +8y +z, +y +z, +y z re consistent nd hve unique solution.

31 ) Show tht the equtions y 8z, +y z, +y +6z re consistent nd hve infinite sets of solution. 6) Test the system of equtions y z, y +z, + y +8z for consistency. 7) Show tht the equtions y, 6 y re inconsistent. 8) Show tht the equtions + y + z, +y z, +y +7z 7 re not consistent. 9) Show tht the equtions +y +z, y z, +y z hve no other solution other thn, y nd z. ) Show tht the equtions +y z, y +z, + 6y z hve non trivil solutions lso. ) Find k if the equtions +y z, yz, +y z k re consistent. ) Find k if the equtions +y +z, y z, +y + z k re inconsistent. ) Find the vlue of k for the equtions y +z, +y z, -y + k z to hve non trivil solutions. ) Find k for which the equtions +y +z, +y +z nd 7 +ky +9z hve no non trivil solutions.. SOLUTION OF LINEAR EQUATIONS.. Solution by Mtri method. When A, the equtions AX B hve the unique solution given by X A - B. Emple Solve using mtrices the equtions -y, +y. The equtions cn be written in mtri form s - y A X B

32 6 A 7 The unique solution is given by X A - B y 7 y y, y Emple Solve the equtions +8y +z, +y +z -, +y -z by using mtri method. The equtions cn be written in mtri form s - 8 z y A X B A - 8 The unique solution is given by X A B. We now find A. A c A t c cofctors +(--), -(--), +(-) -(-8-), +(--), -(-8) +(8-), -(-), +(-8)

33 Now A - y z A A t c y ie., y z z, y, z. Emple 6 6 A womn invested different mounts t 8%, 8 % nd 9%, ll t simple interest. Altogether she invested Rs., nd erns Rs., per yer. How much does she hve invested t ech rte if she hs Rs., more invested t 9% thn t 8%? Solve by using mtrices. Let, y, z be the mounts in Rs. invested t 8%, 8 % nd 9% respectively. According to the problem, + y + z, 8 y z z, + y + z, +y + 6z,8, z, The equtions cn be written in mtri form s, nd

34 Now, Now, 6 y,8, z, A X B A 6 The unique solution is given by X A B We now find A. A c A t c 68 A - y z y z y 68 A A t c , 8,,,,, cofctors +(--), -(--6), +(-) -(--), +(--), -(-) +(6-), -(6-), +(-) 8,,8,, z Hence the mounts invested t 8%, 8 % nd 9% re Rs.,, Rs., nd Rs., respectively.

35 .. Solution by Determinnt method (Crmer s rule) Let the equtions be + b y + c z d, + b y + c z d, + b y + c z d. Let y b b b d d d c c c c c c,, z When, the unique solution is given by, y y, z z. Emple 7 Solve the equtions +y +z, +y +z 6, 6 +y +7z 7 by determinnt method. The equtions re +y +z +y +z 6 6 +y +7z y By Crmer's rule 9 d d d 6 ; ; z b b b b b b c c c d d d

36 z z 6 ; y y 6 8 ;, y, z. 6 Emple 8 Solve the equtions -y-, +y - by Crmer s rule. The equtions re y, +y 9 ; y 9 ; By Crmer's rule 8, y y 9 9 9, y. 8 Emple 9 A slesmn hs the following record of sles during three months for three items A, B nd C which hve different rtes of commission. Months Sles of units Totl commission A B C drwn (in Rs.) Jnury 9 8 Februry 9 Mrch 6 8 Find out the rtes of commission on the items A, B nd C. Solve by Crmer s rule. Let, y nd z be the rtes of commission in Rs. per unit for A, B nd C items respectively.

37 According to the problem, 9 +y +z 8 +y +z 9 6 +y +z 8 Dividing ech of the equtions by throughout, 9 +y + z 8 + y + z y + z 8 Now, ; y By Crmer's rule -7 ; z ; y y z z 9 7 Hence the rtes of commission for A, B nd C re Rs., Rs. nd Rs. respectively. EXERCISE. ) Solve by mtri method the equtions +y 7, + y. ) Solve by mtri method the equtions y +z, y +z, +y z ) Solve by Crmer s rule the equtions 6-7y 6, 9 y. ) Solve by determinnt method the equtions +y z, + y z, +y z. ) Solve by Crmer s rule : + y, y + z 6, z +.

38 6) Two types of rdio vlves A, B re vilble nd two types of rdios nd Q re ssembled in smll fctory. The fctory uses vlves of type A nd vlves of type B for the type of rdio, nd for the rdio Q it uses vlves of type A nd vlves of type B. If the number of vlves of type A nd B used by the fctory re nd 8 respectively, find out the number of rdios ssembled. Use mtri method. 7) The cost of kg. of whet nd kg. of sugr is Rs. 7. The cost of kg. whet nd kg. of rice is Rs. 7. The cost of kg. of whet, kg. of sugr nd kg. of rice is Rs. 7. Find the cost of ech per kg., using mtri method. 8) There re three commodities X, Y nd Z which re bought nd sold by three delers A, B ndc. Deler A purchses units of X nd units of Z nd sells units of Y, deler B purchses units of X, units of Y nd sells 7 units of Z nd deler C purchses units of Y, unit of Z nd sells units of X. In the process A erns Rs. nd C Rs. but B loses Rs.. Find the price of ech of the commodities X, Y nd Z, by using determinnts. 9) A compny produces three products everydy. The totl production on certin dy is tons. It is found tht the production of the third product eceeds the production of the first product by 8 tons while the totl production of the first nd third product is twice the production of second product. Determine the production level of ech product by using Crmer s rule.. STORING INFORMATION We know tht mtri provides convenient nd compct nottion for representtion of dt which is cpble of horizontl nd verticl divisons. Now we shll study the pplictions of mtrices in the study of (i) Reltions on sets (ii) Directed routes nd (iii) Cryptogrphy. Let us first recll the concept of reltions on sets studied in erlier clsses.

39 Reltion : A reltion R from set A to set B is subset of the crtesin product A B. Thus R is set of ordered pirs where the first element comes from A nd the second element comes from B. If (, b) R we sy tht is relted to b nd write R b. If (, b) R, we sy tht is not relted to b nd write R b. If R is reltion from set A to itself then we sy tht R is reltion on A. For emple, Let A {,,, 6} nd B {, 6, 9} Let R be the reltion from A to B defined by Ry if divides y ectly. Then R {(, ), (, 6), (, 6), (, 9), (, ), (6, 6)} Inverse reltion. Let R be ny reltion from set A to set B. Then the inverse of R, denoted by R - is the reltion from B to A which consists of those ordered pirs which, when reversed, belong to R. For emple, the inverse of the reltion R {(, y) (, z) (, y)} from A {,, } to B {, y, z} is R - {(y, ) (z, ) (y, )} from B to A. Composition of reltions. Let A, B nd C be sets nd let R be reltion from A to B nd let S be reltion from B to C. i.e. R is subset of A B nd S is subset of B C. Then R nd S give rise to reltion from A to C denoted by R o S nd defined by R o S {(, c) / there eists b B for which (, b) R nd (b, c) S}. For emple, Let A {,,, }, B {, b, c, d}nd C {, y, z} nd let R {(, ), (, d), (, ), (, b), (, d)} nd S {(b, ), (b, z), (c, y), (d, z)} Then R o S {(, z), (, ), (, z)}

40 Types of reltions. A reltion R on set A is refleive if R for every A. tht is (, ) R for every A. A reltion R on set A is symmetric if whenever Rb then br tht is, whenver (, b) R then (b, ) R. A reltion R on set A is trnsitive if whenever Rb nd brc then Rc tht is, whenever (, b), (b, c) R then (, c) R. A reltion R is n equivlence reltion if R is refleive, symmetric nd trnsitive. For emple, consider the following three reltions on A {,, }. R {(, ), (, ), (, ), (, )} S {(, ), (, ), (, ), (, ), (, )} T {(, ), (, ), (, ), (, )} R is not refleive, S is refleive nd T is not refleive. R is not symmetric, S is symmetric nd T is not symmetric. R is trnsitive, S is trnsitive nd T is not trnsitive... Reltion mtrices. A mtri is convenient wy to represent reltion R from X to Y. Such reltion cn be nlysed by using computer. We lbel the rows with the elements of X (in some rbitry order) nd we lbel the columns with the elements of Y (gin in some rbitry order). We then set the entry in row nd column y to if R y nd to otherwise. The mtri so obtined is clled the reltion mtri for R. Emple Find the reltion mtri for the reltion R from {,, } to {, 6, 7, 8} where R is defined by Ry if divides y ectly. R {(, 6), (, 8), (, 6), (, 8)}

41 The reltion mri for R is R Emple Let S {,,, }. Let R be the reltion on S defined by mrn if m < n. Write out the reltion mtri for R. R {(, ), (, ), (, ), (, ), (, ), (, )} The reltion mtri for R is R Emple Given reltion mtri R y reltion R in the form of set of ordered pirs. R {(, ), (y, ), (y, )} Write down the Mtri for inverse reltion If R is reltion mtri, then its trnspose R t represents the inverse reltion R -. Emple Let A {,, } Define reltion R on A by mrn if mn m. Find the reltion mtri for R. Use it to find the reltion mtri for the inverse reltion R -. R {(, ), (, ), (, ), (, ), (, )}

42 6 Reltion mtri for R is R Reltion mtri for R - is R - R t Reltion mtri for composition of reltions. The reltion mtri for R o R is obtined by replcing ech non zero element in the mtri product R R by. Emple Let R be reltion from X {,, } to Y {, b, c} defined by R {(, ), (, b), (, ), (, b)} nd let R be the re ltion from Y to Z {, y, z} defined by R {(, ), (, y), (b, y), (b, z)}. Find the reltion mtrices for R nd R nd using them find the reltion mtri for R o R. The reltion mtri for R is R c b The reltion mtri for R is R c b z y The mtri product R R is

43 7 R R Replcing ech non zero element in R R by we get, R o R z y Type of reltion s reveled by its mtri. A reltion R is refleive if its mtri hs only s on the min digonl. A reltion R is symmetric if its mtri is symmetric Tht is, i j j i for ll i, j. A reltion R is trnsitive if whenever the entry i, j in the mtri product R is non zero, the entry i, j in the reltion mtri R is lso non zero. Emple Given reltion R {(, ), (b, b), (c, c), (d, d), (b, c), (c, b)} on A {, b, c, d}. Find the reltion mtri for R nd using it identify the type of the reltion. The reltion mtri for R is R d c b d c b The mtri hs only s on the min digonl. Hence the reltion is refleive. The mtri is symmetric. Hence the reltion is symmetric. The mtri product.

44 8 R Whenever n entry in R is non zero, the corresponding entry in R is lso non zero. Hence the reltion is trnsitive. Thus the reltion R is refleive, symmetric nd trnsitive nd hence n equivlence reltion. Emple 6 Let R be the reltion on S {,,, } defined by mrn if m-n <. Find the reltion mtri for R nd using it, identify the type of the reltion. R {(, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, )} The reltion mtri for R is R The mtri hs only s on the min digonl. Hence the reltion is refleive. The mtri is symmetric. Hence the reltion is symmetric. R Now element in R is non zero but element in R is zero. Hence the reltion is not trnsitive.

45 .. Route Mtrices. A directed route is set of points,,..., n clled vertices together with finite set of directed edges ech of which joins n ordered pir of distinct vertices. Thus the directed edge ij is different from the directed edge ji. There my be no directed edge from verte i to ny other verte nd my not be ny directed edge from ny verte to the verte i. Also there cn be no loops nd multiple directed edges joining ny two vertices. Ech edge of directed route is clled stge of length. A pth from the verte i to the verte j is sequence of directed edges from i to j. It should strt from i nd end with j, repetition of ny of the vertices including i nd j being llowed. If there is pth from i to j we sy tht j is ccessible from i or tht i hs ccess to j. A directed route is sid to be strongly connected if for ny pir of vertices i nd j, there is pth from i to j nd pth from j to i. Otherwise the route is sid to be not strongly connected. A directed route is represented by its route mtri. If G is directed route with n vertices then the n n mtri A where the (i, j) th element is if there is directed edge from i to j nd zero otherwise is clled the route mtri for the directed route G. It is to be noted tht the number of s in the route mtri will be equl to the number of edges in its route. Route mtrices re lso reltion mtrices. The route mtrices re squre mtrices wheres the reltion mtrices need not be squre mtrices. th mtri of directed route is the mtri { ij } such tht ij Emple 7, if there is pth from i to j, otherwise. Find the route mtri for ech directed route given below: 9

46 (i) (ii) (iii) (iv) (i) (ii) (iii) (iv) Emple 8 Drw the directed route for ech route mtri given below : (i) (ii) 6 6. Fig.. Fig.. Fig..6 Fig..7

47 (i) (ii) Fig..8 Fig..9 6 Theorems on directed routes. Theorem Si theorems (without proof) re stted below. If A is the route mtri then the i, j th element in A r is the number of wys in which i hs ccess to j in r stges Theorem If A is the route mtri then the sum of ll the elements in the j th column of A r is the number of wys in which j is ccessible by ll individuls in r stges. Theorem If A is the route mtri then the i, j th element in A + A + A A r is the number of wys in which i hs ccess to j in one, two,... or r stges. Theorem If A is the route mtri then the sum of ll the elements in the j th column in A + A + A A r is the number of wys in which J is ccessible by ll individuls in one, two,... or r stges. Theorem A directed route with n vertices nd hving the route mtri A is strongly connected if A + A + A A n hs no zero entries.

48 Theorem 6 If there re n vertices in route nd A is its route mtri then its pth mtri is got by replcing ech non zero element in A + A + A A n by. Now we shll hve few emples illustrting the ppliction of these theorems. Emple 9 Consider the following directed route G. 6 (i) Find the route mtri of G. (ii) (iii) (iv) (v) (vi) Find the number of wys in which hs ccess to in stges. Indicte the pths. Find the number of pths from to in stges Indicte the pths. Find the number of wys in which 6 cn be ccessed by others in stges. Find the number of wys in which hs ccess to in one, two or three stges. Find the number of wys in which 6 cn be ccessed by others in or less stges. (i) Route mtri of G is Fig..

49 A 6 6 (ii) A 6 6 A 6 6 hs ccess to in stges in wys. The pths re,, 6, 6, nd 6. (iii) The number of pths from to in stges is. The pths re, 6,, 6, 6 nd (iv) The number of wys in which 6 cn be ccessed by others in stges (v) A + A + A hs ccess to in one, two, or three stges in 7 wys.

50 (vi) The number of wys in which 6 cn be ccessed by others in or less stges Emple Show, by using the route mtri nd its powers, tht the directed route G given below is strongly connected. The route mtri of G is A Since there re vertices, let us find A + A + A +A. A, A A, A + A + A +A 6 There is no zero entry in this mtri G is strongly connected. Fig..

51 Emple Given directed route : Find the route mtri of G nd using its powers emine whether G is strongly connected. Route mtri of G is A Since there re vertices, let us find A + A + A + A. A A A A +A +A +A There is zero entry in this mtri. Hence G is not strongly connected. Fig..

52 6 Emple Show tht the directed route G with route mtri A is strongly connected. Solution: Since there re vertices, we find A + A + A. A, A. A + A + A There is no zero entry in this mtri. G is strongly connected. Emple Find the pth mtri of the directed route given below by using the powers of its route mtri. The route mtri is A Since there re vertices, let us find A + A + A + A. Fig..

53 7 A, A, A A+A +A + A Replcing ech non zero entry by, we get the pth mtri Emple The route mtri of directed route G is A Find the pth mtri of G without using the powers of A. The directed route G is Fig..

54 The pth mtri is written directly from G... Cryptogrphy Cryptogrphy is the study of coding nd decoding secret messges. A non singulr mtri cn be effectively used for this. The following emple illustrtes the method. Emple Using the substitution scheme, A B C D E F G H I J K L M N O Q R S T U W X Y Z nd the mtri A (i) Code the messge : HARD WORK nd (ii) Decode the messge : 98, 9,, 9, 8, (i) Using the substitution scheme, H A R D W O R K Grouping them, 8 8 8,,, 8

55 Applying the trnsformtion AX B, The coded messge is, 7,,, 6, 6,, 7 (ii) 98, 9,, 9, 8, Grouping them, 98 8,, 9 9 Now solve for AX B. X A B, where A Hence we hve ,,,,, 9 9 Using the substitution scheme, the messge decoded is S A E M E

56 Note When non singulr mtri is used, we group the numbers in twos in order. If number is left out without being pired, we cn include one etrneous number of our own s the lst number nd ignore its decoding, when the process is over. When mtri is used we group the numbers in threes in order. If need be one or two etrneous numbers my be included nd dispensed with when the process is over. EXERCISE. ) Find the reltion mtri for the reltion R from {,, 8, 9} to {6, 8, 9, } defined by R y if divides y ectly. ) Let S {,, 6, 9} nd R be the reltion on S defined by m R n if m > n. Write out the reltion mtri for R. l m ) Given the reltion mtri R b c Write the reltion R in the form of set of ordered pirs. ) Given the reltion mtri R b c d b Write down the mtri for the inverse reltion R. ) Let R be the reltion from X {,, 9} to Y {,, 8} defined by Ry if + y >. Let S be the reltion from Y to Z {,, } defined by ysz if y < z. Find the reltion mtrices for R, S nd Ro S. 6) Find the reltion mtri for the reltion. R {(, ), (, ), (, ), (, )} on {,,, }. Use it to identify the type of the reltion. 7) Find the reltion mtri for the reltion R {(, ), (, ), (, ), (, ), (, )} on {,,, }. Using it, decide the type of the reltion.

57 8) Find the reltion mtri for the reltion R {(, ), (, ), (, ), (, )} on {,,, }. Using it, decide the nture of the reltion. 9) Find the reltion mtri for the reltion R {(, ), (, )} on {,,, }. Using it, identify the type of the reltion. ) Find the route mtri for ech of the directed routes : (i) (ii) (iii) Fig.. Fig..6 (iv) Fig..7 Fig..8 (v) (vi) 6 Fig..9 Fig.. ) Drw the directed route for ech of the following route mtrices (i) A (ii)

58 A B C D A B ) Given the route mtri M C D for directed route G. Using the powers of M, find the number of pths from C to A with t most three stges. Indicte the pths. ) Given the directed route G : X Y (i) Find the route mtri of G. (ii) Z Fig.. Find whether G is strongly connected, by using the powers of the route mtir. (iii) Find the pth mtri of G. ) Given the following directed route G : W (i) Find the route mtri of G. (ii) Find the number of pths of length from to. Indicte them. (iii) Find the number of pths of length from to. Indicte them. (iv) Fig.. Find the number of pths from to of length or less. Indicte them.

59 (v) (vi) Find the number of wys in which cn be ccessed by others in one, two or three stges. Is G strongly connected? (vii) Find the pth mtri of G. ) Given directed route G : Fig.. Find the route mtri of G nd using its powers show tht G is not strongly connected. 6) Given the route mtri for directed route G : Show tht G is strongly connected. 7) Given the directed route G : Fig.. Find the route mtri nd using its powers, find the pth mtri. 8) The route mtri of directed route G is A Find its pth mtri without using the powers of A.

60 9) Given directed route G : Fig.. Find its route nd pth mtrices. ) Using the substitution scheme, A B C D E F G H I J K L M N O Q R S T U W X Y Z nd the mtri A (i) Code the messge : CONSUMER nd (ii) Decode the messge : 68, 8, 8, 6, 6,, 8, 7.. INUT - OUTUT ANALYSIS Consider simple economic model consisting of two industries A nd A, where ech produces only one type of product. Assume tht ech industry consumes prt of its own output nd rest from the other industry for its opertion. The industries re thus interdependent. Further ssume tht whtever is produced is consumed. Tht is the totl output of ech industry must be such s to meet its own demnd, the demnd of the other industry nd the eternl demnd tht is the finl demnd. Our im is to determine the output levels of ech of the two industries in order to meet chnge in finl demnd, bsed on knowledge of the current outputs of the two industries, of course under the ssumption tht the structure of the economy does not chnge.

61 Let i j be the rupee vlue of the output of A i consumed by A j, i, j, Let nd be the rupee vlue of the current outputs of A nd A respectively. Let d nd d be the rupee vlue of the finl demnds for the outputs of A nd A respectively. These ssumptions led us to frme the two equtions Tht is + + d + + d () } Let b ij ij j, i, j, b, b, b Then equtions () tke the form b + b + d b + b + d These cn be rerrnged s (b ) b d, b, b + (b ) d This tkes the mtri form b b d b b d Tht is (I B) X D b Where B b Solving this X (I B) - D. b b d, X nd D d The mtri B is known s the technology mtri.

62 Hwkins - Simon conditions ensure the vibility of the system. If B is the technology mtri then Hwkins Simon conditions re (i) the min digonl elements in I B must be positive nd (ii) I B must be positive. Emple 6 The dt below re bout n economy of two industries nd Q. The vlues re in lkhs of rupees. roducer User Finl Demnd Totl Output Q 6 Q 8 Find the technology mtri nd test whether the system is vible s per Hwkins - Simon conditions. With the usul nottion we hve, 6,,, 8, Now b 6, b, b, b 8. The technology mtri is B I - B The min digonl elements in I B viz., nd re positive. Also 6

63 I B. I B is positive. The two Hwkins - Simon conditions re stisfied. Hence the system is vible. Emple 7 In n economy there re two industries nd Q nd the following tble gives the supply nd demnd positions in crores of rupees. roducer User Finl Demnd Totl Output Q Q 6 Determine the outputs when the finl demnd chnges to for nd for Q. With the usul nottion we hve,,, 6 Now, b, b b, b The technology mtri is B I - B I - B 7 7, 6. 6

64 Now, (I - B) - 7 X (IB) D The output of the industry should be Rs. crores nd tht of Q should be Rs. crores. EXERCISE. ) The technology mtri of n economic system of two industries is. Test whether the system is vible s per Hwkins Simon conditions. ) The technology mtri of n economic system of two industries 9 is. Test whether the system is vible s per Hwkins Simon conditions. ) The technology mtri of n economic system of two industries is 7. Find the output levels when the finl demnd chnges to nd units. ) The dt below re bout n economy of two industries nd Q. The vlues re in millions of rupees. roducer User Finl Demnd Totl Output Q Q Determine the outputs if the finl demnd chnges to for nd for Q.

65 ) Suppose the inter-reltionship between the production of two industries nd Q in yer (in lkhs of rupees) is roducer User Finl Demnd Totl Output Q Q 6 Find the outputs when the finl demnd chnges to (i) for nd 8 for Q (ii) 8 for nd for Q. 6) In n economy of two industries nd Q the following tble gives the supply nd demnd positions in millions of rupees. roducer User Finl Demnd Totl Output Q 6 Q 8 8 Find the outputs when the finl demnd chnges to 8 for nd for Q. 7) The dt below re bout n economy of two industries nd Q. The vlues re in crores of rupees. roducer User Finl Demnd Totl Output Q 7 7 Q Find the outputs when the finl demnd chnges to for nd 6 for Q. 8) The inter - reltionship between the production of two industries nd Q in crores of rupees is given below. roducer User Totl Output Q 8, Q 6, 9

66 If the level of finl demnd for the output of the two industries is, for nd, for Q, t wht level of output should the two industries operte?.6 TRANSITION ROBABILITY MATRICES These re mtrices in which the individul elements re the probbilities of trnsition from one stte to nother of n event. The probbilities of the vrious chnges pplied to the initil stte by mtri multipliction gives forecst of the succeeding stte. The following emples illustrte the method. Emple 8 Two products A nd B currently shre the mrket with shres 6% nd % ech respectively. Ech week some brnd switching tkes plce. Of those who bought A the previous week, 7% buy it gin wheres % switch over to B. Of those who bought B the previous week, 8% buy it gin wheres % switch over to A. Find their shres fter one week nd fter two weeks. If the price wr continues, when is the equilibrium reched? Trnsition robbility mtri A B A T.7. B..8 Shres fter one week A B A B A.7. A B (.6. ) (.. ) B..8 A %, B % Shres fter two weeks A B A B A.7. A (.. ). B..8 A %, B % (.) 6 B

67 Equilibrium At equilibrium we must hve (A B) T (A B) where A + B.7. (A B)..8 (A B).7 A +. B A.7 A +. (-A) A Simplifying, we get A. Equilibrium is reched when A s shre is % nd B s shre is 6% Emple 9 A new trnsit system hs just gone into opertion in city. Of those who use the trnsit system this yer, % will switch over to using their own cr net yer nd 9% will continue to use the trnsit system. Of those who use their crs this yer, 8% will continue to use their crs net yer nd % will switch over to the trnsit system. Suppose the popultion of the city remins constnt nd tht % of the commuters use the trnsit system nd % of the commuters use their own cr this yer, (i) wht percent of commuters will be using the trnsit system fter one yer? (ii) wht percent of commuters will be using the trnsit system in the long run? Trnsition robbility Mtri S C S T.9. C..8 ercentge fter one yer S C S C S.9. A (.. ). C..8 (.) 6 B

68 S %, C % Equilibrium will be reched in the long run. At equilibrium we must hve (S C) T (S C) where S + C (S C) S +.C S.9S +.(-S) S (S C) Simplifying, we get S.67 67% of the commuters will be using the trnsit system in the long run. EXERCISE.6 ) Two products nd Q shre the mrket currently with shres 7% nd % ech respectively. Ech week some brnd switching tkes plce. Of those who bought the previous week, 8% buy it gin wheres % switch over to Q. Of those who bought Q the previous week, % buy it gin wheres 6% switch over to. Find their shres fter two weeks. If the price wr continues, when is the equilibrium reched? ) The subscription deprtment of mgzine sends out letter to lrge miling list inviting subscriptions for the mgzine. Some of the people receiving this letter lredy subscribe to the mgzine while others do not. From this miling list, 6% of those who lredy subscribe will subscribe gin while % of those who do not now subscribe will subscribe. On the lst letter it ws found tht % of those receiving it ordered subscription. Wht percent of those receiving the current letter cn be epected to order subscription? ) Two newsppers A nd B re published in city. Their present mrket shres re % for A nd 8% for B. Of those who bought A the previous yer, 6% continue to buy it gin while % switch over to B. Of those who bought B the previous yer, % buy it gin nd % switch over to A. Find their mrket shres fter two yers. 6

69 6 EXERCISE.7 Choose the correct nswer ) If the minor of equls the cofctor of in i j then the minor of is () (b) (c) (d) ) The Adjoint of is () (b) (c) (d) ) The Adjoint of is () (b) (c) (d) ) If AB BA A I then the mtri B is () the inverse of A (b) the trnspose of A (c) the Adjoint of A (d) A ) If A is squre mtri of order then AdjA is () A (b) A (c) A (d) A 6) If A then AdjA is () (b) (c) (d) + 7) The inverse of is () (b) (c) (d)

70 .8.6 8) If A then A is () (b) (c) (d) ) For wht vlue of k the mtri A, k where A hs no inverse? () (b) (c) (d) ) If A then A A is 7 () (b) A (c) I (d) A. ) The rnk of n n n mtri ech of whose elements is is () (b) (c) n (d) n ) The rnk of n n n mtri ech of whose elements is is () (b) (c) n (d) n ) The rnk of zero mtri is () (b) (c) (d) ) The rnk of non singulr mtri of order n n is () n (b) n (c) (d) ) A system of liner homogeneous equtions hs t lest () one solution (b) two solutions (c) three solutions (d) four solutions 6) The equtions AX B cn be solved by Crmer s rule only when () A (b) A (c) A B (d) A B y 7) The inverse of the reltion is () y b (b) b b y 6 (c) y b (d) b y

71 b 8) The reltion R () Refleive (c) Trnsititve b is (b) Symmetric (d) Refleive nd symmetric 9) The number of Hwkins - Simon conditions for the vibility of n input - output model is () (b) (c) (d) A A.7 B. ) If T is trnsition probbility mtri, then the B.8 vlue of is (). (b). (c). (d).7 6

72 ANALYTICAL GEOMETRY. CONICS Intersections of cone by plne The prbol, ellipse nd hyperbol re ll members of clss of curves clled conics. The bove three curves cn be obtined by cutting cone with plne nd so they re clled conics (Fig..). rbol Circle Hyperbol Ellipse Fig.. A conic is the locus of point which moves in plne such tht its distnce from fied point in the plne bers constnt rtio to its distnce from fied stright line in tht plne. Focus, Directri, Eccentricity: In the bove definition of conic, the fied point is clled the focus, the fied line the directri nd the constnt rtio, the eccentricity of the conic. The eccentricity is usully denoted by the letter e. 66

73 In the Fig.., S is the focus, the line LM is the directri nd S e M M The conic is S (focus) prbol if e, L n ellipse if e < nd hyperbol if e >. directri.. The generl eqution of conic Fig.. We know tht conic is the locus of point moving such tht its distnce from the focus bers constnt rtio to its distnce from the directri. Let the focus be S(, y ) nd the directri be A + By +C Let the eccentricity of the conic be e nd (, y) be ny point on it. Then S ( ) ( ) + y y erpendiculr distnce of from A + By + C is A + By + C M + A + B S e M ( ) + ( y y) A+ By+ C e ± A + B ( A+ By + C) or ( ) + (y y ) e ( A + B ) Simplifying, we get n eqution of the second degree in nd y of the form + hy + by + g + fy + c This is the generl eqution of conic. Remrks : + hy + by + g + fy + c represents, (i) pir of stright lines if bc + fgh f bg ch 67