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1、1.1SOLUTIONSNotes:Thekeyexercisesare7(or11or12),19–22,and25.Forbrevity,thesymbolsR1,R2,…,standforrow1(orequation1),row2(orequation2),andsoon.Additionalnotesareattheendofthesection.xx12+=57??1571.????=275xx12??????275xx12+=57?157?ReplaceR2byR2+(2)R1andobtain:??39x2=?039?xx12+=57?157?ScaleR2
2、by1/3:??x2=3?013?x1=?8?108??ReplaceR1byR1+(–5)R2:??x2=3?013?Thesolutionis(x1,x2)=(–8,3),orsimply(–8,3).244xx12+=??244??2.??571xx12+=1?5711?xx12+=22??122??ScaleR1by1/2andobtain:??571xx12+=1?5711?xx12+=22??122??ReplaceR2byR2+(–5)R1:???=32x21?032?1?xx12+=22??122??ScaleR2by–1/3:??x2=?7?017??x1
3、=12?1012?ReplaceR1byR1+(–2)R2:??x2=?7?017??Thesolutionis(x1,x2)=(12,–7),orsimply(12,–7).12CHAPTER1?LinearEquationsinLinearAlgebra3.Thepointofintersectionsatisfiesthesystemoftwolinearequations:xx12+=57?157???xx12?=22??122???xx12+=57?157?ReplaceR2byR2+(–1)R1andobtain:???=79x2??079???xx12+=57
4、?157?ScaleR2by–1/7:??x2=9/7?019/7?x1=4/7?104/7?ReplaceR1byR1+(–5)R2:??x2=9/7?019/7?Thepointofintersectionis(x1,x2)=(4/7,9/7).4.Thepointofintersectionsatisfiesthesystemoftwolinearequations:xx12?=51??151???375xx12?=??375?xx12?=51?151??ReplaceR2byR2+(–3)R1andobtain:??82x2=?082?xx12?=51?151??S
5、caleR2by1/8:??x2=1/4?011/4?x1=9/4?109/4?ReplaceR1byR1+(5)R2:??x2=1/4?011/4?Thepointofintersectionis(x1,x2)=(9/4,1/4).5.Thesystemisalreadyin“triangular”form.Thefourthequationisx4=–5,andtheotherequationsdonotcontainthevariablex4.Thenexttwostepsshouldbetousethevariablex3inthethirdequationtoel
6、iminatethatvariablefromthefirsttwoequations.Inmatrixnotation,thatmeanstoreplaceR2byitssumwith3timesR3,andthenreplaceR1byitssumwith–5timesR3.6.Onemorestepwillputthesystemintriangularform.ReplaceR4byitssumwith–3timesR3,which??16401????02704?produces??.Afterthat,thenextstepistoscalethefourthr
7、owby–1/5.??00123?????00051?57.Ordinarily,thenextstepwouldbetointerchangeR3andR4,toputa1inthethirdrowandthirdcolumn.Butinthiscase,thethirdrowoftheaugmentedmatrixcorrespondstotheequation0x1+0x2+0x3=1,orsimply,0=1.Asystemcontainingthisconditionhasnosolution.Furth
