gauss-jordan

Gauss-JordanEliminationMethodThefollowingrowoperationsontheaugmentedmatrixofasystemproducetheaugmentedmatrixofanequivalentsystem,i.e.,asystemwiththesamesolutionastheoriginalone.•Interchangeanytworows.•Multiplyeachelementofarowbyanonzeroconstant.•Replacearowbythesumofitselfandaconstantmultipleofanotherrowofthematrix.Fortheserowoperations,wewillusethefollowingnotations.•Ri↔Rjmeans:Interchangerowiandrowj.•αRimeans:Replacerowiwithαtimesrowi.•Ri+αRjmeans:Replacerowiwiththesumofrowiandαtimesrowj.TheGauss-Jordaneliminationmethodtosolveasystemoflinearequationsisdescribedinthefollowingsteps.1.Writetheaugmentedmatrixofthesystem.2.Userowoperationstotransformtheaugmentedmatrixintheformdescribedbelow,whichiscalledthereducedrowechelonform(RREF).(a)Therows(ifany)consistingentirelyofzerosaregroupedtogetheratthebottomofthematrix.(b)Ineachrowthatdoesnotconsistentirelyofzeros,theleftmostnonzeroelementisa1(calledaleading1orapivot).(c)Eachcolumnthatcontainsaleading1haszerosinallotherentries.(d)Theleading1inanyrowistotheleftofanyleading1’sintherowsbelowit.3.Stopprocessinstep2ifyouobtainarowwhoseelementsareallzerosexceptthelastoneontheright.Inthatcase,thesystemisinconsistentandhasnosolutions.Otherwise,finishstep2andreadthesolutionsofthesystemfromthefinalmatrix.Note:Whendoingstep2,rowoperationscanbeperformedinanyorder.Trytochooserowopera-tionssothatasfewfractionsaspossiblearecarriedthroughthecomputation.Thismakescalculationeasierwhenworkingbyhand.1Example1.SolvethefollowingsystembyusingtheGauss-Jordaneliminationmethod.x+y+z=52x+3y+5z=84x+5z=2Solution:Theaugmentedmatrixofthesystemisthefollowing.111523584052Wewillnowperformrowoperationsuntilweobtainamatrixinreducedrowechelonform.111523584052R2−2R1−−−−−→1115013−24052R3−4R1−−−−−→1115013−20−41−18R3+4R2−−−−−→1115013−20013−26113R3−−−→1115013−2001−2R2−3R3−−−−−→11150104001−2R1−R3−−−−→11070104001−2R1−R2−−−−→10030104001−2Fromthisfinalmatrix,wecanreadthesolutionofthesystem.Itisx=3,y=4,z=−2.2Example2.SolvethefollowingsystembyusingtheGauss-Jordaneliminationmethod.x+2y−3z=26x+3y−9z=67x+14y−21z=13Solution:Theaugmentedmatrixofthesystemisthefollowing.12−3263−96714−2113Let’snowperformrowoperationsonthisaugmentedmatrix.12−3263−96714−2113R2−6R1−−−−−→12−320−99−6714−2113R3−7R1−−−−−→12−320−99−6000−1Weobtainarowwhoseelementsareallzerosexceptthelastoneontheright.Therefore,weconcludethatthesystemofequationsisinconsistent,i.e.,ithasnosolutions.Example3.SolvethefollowingsystembyusingtheGauss-Jordaneliminationmethod.4y+z=22x+6y−2z=34x+8y−5z=4Solution:Theaugmentedmatrixofthesystemisthefollowing.041226−2348−543Wewillnowperformrowoperationsuntilweobtainamatrixinreducedrowechelonform.041226−2348−54R1↔R2−−−−−→26−23041248−54R3−2R1−−−−−→26−2304120−4−1−2R3+R2−−−−→26−230412000014R2−−−→26−23011/41/20000R1−6R2−−−−−→20−7/20011/41/2000012R1−−−→10−7/40011/41/20000Thislastmatrixisinreducedrowechelonformsowecanstop.Itcorrespondstotheaugmentedmatrixofthefollowingsystem.{x−74z=0y+14z=12Wecanexpressthesolutionsofthissystemasx=74z,y=12−14z.Sincethereisnospecificvalueforz,itcanbechosenarbitrarily.Thismeansthatthereareinfinitelymanysolutionsforthissystem.Wecanrepresentallthesolutionsbyusingaparametertasfollows.x=74t,y=12−14t,z=tAnyvalueoftheparametertgivesusasolutionofthesystem.Forexample,t=4givesthesolution(x,y,z)=(7,−12,4)t=−2givesthesolution(x,y,z)=(−72,1,−2).4Example4.SolvethefollowingsystembyusingtheGauss-Jordaneliminationmethod.A+B+2C=12A−B+D=−2A−B−C−2D=42A−B+2C−D=0Solution:Wewillperformrowoperationsontheaugmentedmatrixofthesystemuntilweobtainamatrixinreducedrowechelonform.112012−101−21−1−1−242−12−10R2−2R1−−−−−→112010−3−41−41−1−1−242−12−10R3−R1−−−−→112010−3−41−40−2−3−232−12−10R4−2R1−−−−−→112010−3−41−40−2−3−230−3−2−1−2R4−R2−−−−→112010−3−41−40−2−3−23002−22R2↔R3−−−−−→112010−2−3−230−3−41−4002−22−12R2−−−−→11201013/21−3/20−3−41−4002−22R3+3R2−−−−−→11201013/21−3/2001/24−17/2002−222R3−−→11201013/21−3/20018−17002−22R4−2R3−−−−−→11201013/21−3/20018−17000−1836−118R4−−−−→11201013/21−3/20018−170001−2R3−8R4−−−−−→11201013/21−3/20010−10001−2R2−R4−−−−→11201013/201/20010−10001−2R2−32R3−−−−−→11201010020010−10001−2R1−2R3,R1−R2−−−−−−−−−−−→10001010020010−10001−2Fromthisfinalmatrix,wecanreadthesolutionofthesystem.ItisA=1,B=2,C=−1,D=−2.5