Further Mathematics for Economic Analysis Student’

Student’sManualFurtherMathematicsforEconomicAnalysis2ndeditionArneStrømKnutSydsæterAtleSeierstadPeterHammondForfurthersupportingresourcespleasevisit:’ssolutionsmanualaccompaniesFurtherMathematicsforEconomicAnalysis(2ndedition,FTPrenticeHall,2008).Itsmainpurposeistoprovidemoredetailedsolutionstotheproblemsmarkedwith⊂SM⊃inthetext.TheManualshouldbeusedinconjunctionwiththeanswersinthebook.Insomefewcasesonlyapartoftheproblemisdoneindetail,becausetherestfollowsthesamepattern.Attheendofthismanualthereisalistofmisprintsandothererrorsinthebook,andevenoneortwointheerratalistinthepreliminaryandincompleteversionofthismanualreleasedinSeptemberthisyear.Wewouldgreatlyappreciatesuggestionsforimprovementsfromourreadersaswellashelpinweedingouttheinaccuraciesanderrorsthatprobablyremain.OsloandCoventry,October2008ArneStrøm([e-mail])KnutSydsæter([e-mail])AtleSeierstad([e-mail])PeterHammond([e-mail])Contents1TopicsinLinearAlgebra............................................................12MultivariableCalculus.............................................................123StaticOptimization...............................................................234TopicsinIntegration..............................................................385DifferentialEquationsI:First-OrderEquationsinOneVariable.............................436DifferentialEquationsII:Second-OrderEquationsandSystemsinthePlane..................497DifferentialEquationsIII:Higher-OrderEquations......................................568CalculusofVariations.............................................................609ControlTheory:BasicTechniques...................................................6510ControlTheorywithManyVariables..................................................8111DifferenceEquations..............................................................9212DiscreteTimeOptimization.........................................................9613TopologyandSeparation..........................................................10214CorrespondencesandFixedPoints..................................................105ASets,Completeness,andConvergence................................................107BTrigonometricFunctions..........................................................109Correctionstothebook...........................................................112Version1.028102008879©ArneStrøm,KnutSydsæter,AtleSeierstad,andPeterHammond2008CHAPTER1TOPICSINLINEARALGEBRA1Chapter1TopicsinLinearAlgebra1.21.2.3LetA=⎛⎝120011101⎞⎠beamatrixwiththethreegivenvectorsascolumns.Cofactorexpansionof|A|alongthefirstrowyields120011101=11101−20111+0=1−2(−1)=3=0ByTheorem1.2.1thisshowsthatthegivenvectorsarelinearlyindependent.1.2.6Part(a)isjustaspecialcaseofpart(b),sowewillonlyprove(b).Toshowthata1,a2,...,anarelinearlyindependentitsufficestoshowthatifc1,c2,...,cnarerealnumberssuchthatc1a1+c2a2+···+cnan=0thenallthecihavetobezero.Sosupposethatwehavesuchasetofrealnumbers.Thenforeachi=1,2,...,n,wehaveai·(c1a1+c2a2+···+cnan)=ai·0=0(1)Sinceai·aj=0wheni=j,theleft-handsideof(1)reducestoai·(ciai)=ciai2.Hence,ciai2=0.Becauseai=0wehaveai=0,anditfollowsthatci=0.1.31.3.1(a)Therankis1.Seetheanswerinthebook.(b)Theminorformedfromthefirsttwocolumnsinthematrixis1320=−6=0.Sincethisminorisoforder2,therankofthematrixmustbeatleast2,andsincethematrixhasonlytworows,therankcannotbegreaterthan2,sotherankequals2.(c)Thefirsttworowsandlasttwocolumnsofthematrixyieldtheminor−13−47=5=0,sotherankofthematrixisatleast2.Ontheotherhand,allthefourminorsoforder3arezero,sotherankislessthan3.Hencetherankis2.(Itcanbeshownthatr2=3r1+r3,wherer1,r2,andr3aretherowsofthematrix.)Analternativeargumentrunsasfollows:Therankofamatrixdoesnotchangeifweaddamultipleofonerowtoanotherrow,so⎛⎝12−1324−47−1−2−1−2⎞⎠−21←←∼⎛⎝12−1300−2100−21⎞⎠.©ArneStrøm,KnutSydsæter,AtleSeierstad,andPeterHammond20082CHAPTER1TOPICSINLINEARALGEBRAHere∼meansthatthelastmatrixisobtainedfromthefirstonebyelementaryrowoperations.Thelastmatrixobviouslyhasrank2,andthereforetheoriginalmatrixalsohasrank2.(d)Thefirstthreecolumnsofthematrixyieldtheminor1302401−12=−4=0,sotherankis3.(e)21−14=9=0,sotherankisatleast2.Allthefourminorsoforder3arezero,sotherankmustbelessthan3.Hencetherankis2.(Thethreerows,r1,r2,andr3,ofthematrixarelinearlydependent,becauser2=−14r1+9r3.)(f)Thedeterminantofthewholematrixiszero,sotherankmustbelessthan4.Ontheotherhand,thefirstthreerowsandthefirstthreecolumnsyieldtheminor1−2−1211−11−1=−7=0sotherankisatleast3.Itfollowsthatthematrixhasrank3.1.3.2(a)Thedeterminantis(x+1)(x−2).Therankis3ifx=−1andx=2.Therankis2ifx=−1orx=2.(a)Bycofactorexpansionalongthefirstrow,thedeterminantofthematrixA=⎛⎝x0x2−2011−1xx−1⎞⎠is|A|=x·(−1)−0·1+(x2−2)·1=x2−x−2=(x+1)(x−2)Ifx=−1andx=2,then|A|=0,sotherankofAequals3.Ifx=−1orx=2,then|A|=0andr(A)≤2.Ontheotherhand,theminorwegetifwestrikeoutthefirstrowandthethird