linear models and matrixalgebra (continued)

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1、ChapterLinearModelsandMatrixAlgebra(Continued)InChap.4,itwasshownthatalinear~equatjonsystem,howeverlarge,maybewritteninaC0mpactmatrixnotation.Furthem)ore,suchanequationsystemcanbesolvedbyfindingtheinverseofth('coeffickotmatrix,providedtheinverseexists.Now

2、wemustaddress(Jur•i:.e1vestothequestion~ofhowtotestfortheexistem:coftheJDVerscandhowtofindthatinverse.Onlyafterwehm·cansweredthesequestionswillitbeposslblt"toapplymatrLxalgebrameaningfullytoeconomicmodels.5.1ConditionsforNonsingularityofaMatrixAgivencoeft

3、icientmatrix.1canhaveaninverse(i...:.,canbe"nonsingular")onlyifitissquare.Aswaspointedoutearlier,however,thesqu,irCties~conditionisnece:;sarybutno!sufficientforthet!xistcnccorllleinverseA-'.Amatrixcanbesquare,butsingular(with•outaninverse)nonetheless.Nece

4、ssaryversusSufficientConditionsTheconceptsof"'necessaryconditioll"and"sufficientcondition"areusedfrequentlyineconomics,ItisimportantthatweunderstandtheirpreeisemeamngsbefofCproceedingfurther,Anecessaryconditionisinthenatureofaprerequjsite:Supposethatastat

5、ementpisHueonly(fanotherstatementqistrue:thenI.{constitutesanecessaryconditionofp,Sym•bolically.weexprc:'i:'ithisabfollows:p,*q(5.1)whichisreadas"ponlyI/q,"oralternatively,"lrp,thenq,"Itisalsologicallycorrecttointerpret(5.I)tomean"pimpliesq."Itmayhappen,o

6、f';OUfse.thatwealsohavep=?It'atthesametime,Thenbothqand)l'arcnecessaryconditionsforp.Example1Ifweletpbethestiltement"apersonisafather"andqbethestatement"apersonismale,"thenthelogicalstatementp0;;;}qapplies,Apersonisafatheronlyifheismale,andtobemaleisanece

7、ssaryconditionforfatherhood,Note,however,thattheconversei$nottrue:82fatherhoodisnotanecessaryconditionformaleness,Chapter5Lillear,4,,,Ie/.I'

8、uevhcnqisnottrue.Tnthii$ca:;e.qissaidtobeasufficientcondition1i)(p.Tho.)truthofq::mnice~toestablishthetruthofp,butItisnotanecessaryconditionforp.Thiscaseisexpressedsymholicallyhy(5.2)vhichisn;ud:"pifq"(withoutthew