SolvingLinearSystems
Section2.1,p.94
2.(a)Possibleanswer:
(b)Possibleanswer:
4.(a)
6.(a)
8.(a)REF(b)RREF(c)N
9.Considerthecolumnsof A whichcontainleadingentriesofnonzerorowsof A.Ifthissetofcolumnsis theentiresetof n columns,then A = In .Otherwisetherearefewerthan n leadingentries,andhence fewerthan n nonzerorowsof A
10.(a) A isrowequivalenttoitself:thesequenceofoperationsistheemptysequence.
(b)EachelementaryrowoperationoftypesI,IIorIIIhasacorrespondinginverseoperationofthe sametypewhich“undoes”theeffectoftheoriginaloperation.Forexample,theinverseofthe operation“add d timesrow r of A torow s of A”is“subtract d timesrow r of A fromrow s of A.”Since B isassumedrowequivalentto A,thereisasequenceofelementaryrowoperations whichgetsfrom A to B .Takethoseoperationsinthereverseorder,andforeachoperationdoits inverse,andthattakes B to A.Thus A isrowequivalentto B
(c)Followtheoperationswhichtake A to B withthosewhichtake B to C
12.(a) ⎡
Section2.2,p.113
2.(a) x = 6 s t, y = s, z = t, w =5.
(b) x = 3, y = 2, z =1.
4.(a) x =5+2t, y =2 t, z = t
(b) x =1, y =2, z =4+ t, w = t.
6.(a) x = 2+ r , y = 1, z =8 2r , x4 = r ,where r isanyrealnumber.
(b) x =1, y = 2 3 , z = 2 3 . (c)Nosolution.
8.(a) x =1 r , y =2, z =1, x4 = r ,where r isanyrealnumber.
(b) x =1 r , y =2+ r , z = 1+ r , x4 = r ,where r isanyrealnumber.
10. x = r 0 ,where r =0.
12. x = ⎡ ⎢ ⎢ ⎢ ⎣ 1 4 r 1 4 r r ⎤
14.(a) a = 2.(b) a = ±2.(c) a =2.
16.(a) a = ±√6.(b) a = ±√6.
18.Theaugmentedmatrixis ab 0 cd 0 .Ifwereducethismatrixtoreducedrowechelonform,wesee thatthelinearsystemhasonlythetrivialsolutionifandonlyif A isrowequivalentto I2 .Nowshow thatthisoccursifandonlyif ad bc =0.If ad bc =0thenatleastoneof a or c is =0,anditisa routinemattertoshowthat A isrowequivalentto I2 .If ad bc =0,thenbycaseconsiderationswe findthat A isrowequivalenttoamatrixthathasaroworcolumnconsistingentirelyofzeros,sothat A isnotrowequivalentto I2
Alternateproof:If ad bc =0,then A isnonsingular,sotheonlysolutionisthetrivialone.If ad bc =0,then ad = bc.If ad =0theneither a or d =0,say a =0.Then bc =0,andeither b or c =0.Inanyofthesecaseswegetanontrivialsolution.If ad =0,then a c = b d ,andthesecond equationisamultipleofthefirstonesoweagainhaveanontrivialsolution.
19.ThishadtobeshowninthefirstproofofExercise18above.IfthealternateproofofExercise18was given,thenExercise19followsfromtheformerbynotingthatthehomogeneoussystem Ax = 0 has onlythetrivialsolutionifandonlyif A isrowequivalentto I2 andthisoccursifandonlyif ad bc =0.
22. a + b + c =0.
24.(a)Change“row”to“column.”
(b)ProceedasintheproofofTheorem2.1,changing“row”to“column.”
25.UsingExercise24(b)wecanassumethatevery m × n matrix A iscolumnequivalenttoamatrixin columnechelonform.Thatis, A iscolumnequivalenttoamatrix B thatsatisfiesthefollowing:
(a)Allcolumnsconsistingentirelyofzeros,ifany,areattherightsideofthematrix.
(b)Thefirstnonzeroentryineachcolumnthatisnotallzerosisa1,calledtheleadingentryofthe column.
(c)Ifthecolumns j and j +1aretwosuccessivecolumnsthatarenotallzeros,thentheleading entryofcolumn j +1isbelowtheleadingentryofcolumn j
Westartwithmatrix B andshowthatitispossibletofindamatrix C thatiscolumnequivalentto B thatsatisfies
(d)Ifarowcontainsaleadingentryofsomecolumnthenallotherentriesinthatrowarezero.
Ifcolumn j of B containsanonzeroelement,thenitsfirst(countingtoptobottom)nonzeroelement isa1.Supposethe1appearsinrow rj .Wecanperformcolumnoperationsoftheform acj + ck for eachofthenonzerocolumns ck of B suchthattheresultingmatrixhasrow rj witha1inthe(rj ,j ) entryandzeroseverywhereelse.Thiscanbedoneforeachcolumnthatcontainsanonzeroentryhence wecanproduceamatrix C satisfying(d).Itfollowsthat C istheuniquematrixinreducedcolumn echelonformandcolumnequivalenttotheoriginalmatrix A
26. 3a b + c =0.
28.ApplyExercise18tothelinearsystemgivenhere.Thecoefficientmatrixis a rd cb r .
HencefromExercise18,wehaveanontrivialsolutionifandonlyif(a r )(b r ) cd =0.
29.(a) A(xp + xh )= Axp + Axh = b + 0 = b (b)Let xp beaparticularsolutionto Ax = b andlet x beanysolutionto Ax = b.Let xh = x xp . Then x = xp + xh = xp +(x xp )and Axh = A(x xp )= Ax Axp = b b = 0.Thus xh is infactasolutionto Ax = 0.
30.(a)3x2 +2(b)2x2 x 1
32. 3 2 x2 x + 1 2 .
34.(a) x =0, y =0 (b) x =5, y = 7
36. r =5, r2 =5.
37.TheGPSreceiverislocatedatthetangentpointwherethetwocirclesintersect.
38.4Fe+3O2 → 2Fe2 O3
40. x = 0 1 4 1 4 i
42.Nosolution.
Section2.3,p.124
1.Theelementarymatrix E whichresultsfrom In byatypeIinterchangeofthe ithand j throwdiffers from In byhaving1’sinthe(i,j )and(j,i)positionsand0’sinthe(i,i)and(j,j )positions.Forthat E , EA hasasits ithrowthe j throwof A andforits j throwthe ithrowof A.
Theelementarymatrix E whichresultsfrom In byatypeIIoperationdiffersfrom In byhaving c =0 inthe(i,i)position.Then EA hasasits ithrow c timesthe ithrowof A.
Theelementarymatrix E whichresultsfrom In byatypeIIIoperationdiffersfrom In byhaving c in the(j,i)position.Then EA hasas j throwthesumofthe j throwof A and c timesthe ithrowof A.
2.(a)
4.(a)Add2timesrow1torow3:
(b)Add2timesrow1torow3:
(c)
Therefore B is theinverseof A.
6.If E1 isanelementarymatrixoftypeIthen E 1 1 = E1 .Let E2 beobtainedfrom In bymultiplying the ithrowof In by c =0.Let E ∗ 2 beobtainedfrom In bymultiplyingthe ithrowof In by 1 c .Then E2 E ∗ 2 = In .Let E3 beobtainedfrom In byadding c timesthe ithrowof In tothe j throwof In .Let E ∗ 3 beobtainedfrom In byadding c timesthe ithrowof In tothe j throwof In .Then E3 E ∗ 3 = In .
8. A 1 = ⎡ ⎢ ⎢ ⎢ ⎣ 1 10 3
14. A isrowequivalentto I3 ;apossibleansweris
16. A = ⎡ ⎢ ⎢ ⎢ ⎣ 3 2 1 1 2 1 2 0 1 2
18.(b)and(c).
20.For a = 1or a =3.
21.ThisfollowsdirectlyfromExercise19ofSection2.1andCorollary2.2.Toshowthat
weproceedasfollows:
23.Thematrices A and B arerowequivalentifandonlyif B = Ek Ek 1 ··· E2 E1 A Let P = Ek Ek 1 ··· E2 E1
24.If A and B arerowequivalentthen B = PA,where P isnonsingular,and A = P 1 B (Exercise23).If A isnonsingularthen B isnonsingular,andconversely.
25.Suppose B issingular.ThenbyTheorem2.9thereexists x = 0 suchthat B x = 0.Then(AB )x = A0 = 0,whichmeansthatthehomogeneoussystem(AB )x = 0 hasanontrivialsolution.Theorem 2.9impliesthat AB issingular,acontradiction.Hence, B isnonsingular.Since A =(AB )B 1 isa productofnonsingularmatrices,itfollowsthat A isnonsingular.
AlternateProof:If AB isnonsingularitfollowsthat AB isrowequivalentto In ,so P (AB )= In .Since P isnonsingular, P = Ek Ek 1 E2 E1 .Then(PA)B = In or(Ek Ek 1 E2 E1 A)B = In .Letting Ek Ek 1 E2 E1 A = C ,wehave CB = In ,whichimpliesthat B isnonsingular.Since PAB = In , A = P 1 B 1 ,so A isnonsingular.
26.Thematrix A isrowequivalentto O ifandonlyif A = PO = O where P isnonsingular.
27.Thematrix A isrowequivalentto B ifandonlyif B = PA,where P isanonsingularmatrix.Now B T = AT P T ,so A isrowequivalentto B ifandonlyif AT iscolumnequivalentto B T
28.If A hasarowofzeros,then A cannotberowequivalentto In ,andsobyCorollary2.2, A issingular. Ifthe j thcolumnof A isthezerocolumn,thenthehomogeneoussystem Ax = 0 hasanontrivial solution,thevector x with1inthe j thentryandzeroselsewhere.ByTheorem2.9, A issingular.
29.(a)No.Let A = 10 00 , B = 00 01 .Then(A + B ) 1 existsbut A 1 and B 1 donot.Even supposingtheyallexist,equalityneednothold.Let A = 1 , B = 2 so(A + B ) 1 = 1 3 = 1 + 1 2 = A 1 + B 1
(b)Yes,for A nonsingularand r =0. (rA) 1 r A 1 = r 1 r A A 1 =1 In = In .
30.Supposethat A isnonsingular.Then Ax = b hasthesolution x = A 1 b forevery n × 1matrix b. Conversely,supposethat Ax = b isconsistentforevery n × 1matrix b.Letting b bethematrices
Letting C bethematrixwhose j thcolumnis xj ,wecanwritethe n systemsin(∗)as AC = In ,since In = e1 e2 ··· en .Hence, A isnonsingular.
31.Weconsiderthecasethat A isnonsingularanduppertriangular.Asimilarargumentcanbegivenfor A lowertriangular.
ByTheorem2.8, A isaproductofelementarymatriceswhicharetheinversesoftheelementary matricesthat“reduce” A to In .Thatis,
Theelementarymatrix Ei willbeuppertriangularsinceitisusedtointroducezerosintotheupper triangularpartof A inthereductionprocess.Theinverseof Ei isanelementarymatrixofthesame typeandalsoanuppertriangularmatrix.Sincetheproductofuppertriangularmatricesisupper triangularandwehave A 1 = Ek ··· E1 weconcludethat A 1 isuppertriangular.
Section2.4,p.129
1.SeetheanswertoExercise4,Section2.1.Whereitmentionsonlyrowoperations,nowread“rowand columnoperations”.
2.(a) I4 0 .(b) I3 .(c) I2 0 00 .(d) I4
4.Allowableequivalenceoperations(“elementaryroworelementarycolumnoperation”)includeinparticularelementaryrowoperations.
5. A and B areequivalentifandonlyif B = Et E2 E1
= P and F1 F2 ··· Fs = Q
6. B = I2 0 00 ;apossibleansweris: B =
8.Suppose A werenonzerobutequivalentto O .Thensomeultimateelementaryroworcolumnoperation musthavetransformedanonzeromatrix Ar intothezeromatrix O .Byconsideringthetypesof elementaryoperationsweseethatthisisimpossible.
9.Replace“row”by“column”andviceversaintheelementaryoperationswhichtransform A into B
10.Possibleanswersare:
(a)
11.If A and B areequivalentthen B = PAQ and A = P 1 BQ 1 .If A isnonsingularthen B isnonsingular, andconversely.
Section2.5,p.136
2. x = ⎡ ⎣ 0 2 3 ⎤ ⎦.
4. x = ⎡ ⎢ ⎢ ⎣ 2 1 0 5 ⎤ ⎥ ⎥
6.
8. L = ⎡ ⎢ ⎢ ⎣ 1000 6100 1210 2321 ⎤ ⎥
=
SupplementaryExercisesforChapter2,p.137
2.(a) a = 4or a =2.
(b)Thesystemhasasolutionforeachvalueof a
4. c +2a 3b =0.
5.(a)Multiplythe j throwof B by 1 k
(b)Interchangethe ithand j throwsof B
(c)Add k timesthe j throwof B toits ithrow.
6.(a)Ifwetransform E1 toreducedrowechelonform,weobtain In .Hence E1 isrowequivalentto In andthusisnonsingular.
(b)Ifwetransform E2 toreducedrowechelonform,weobtain In .Hence E2 isrowequivalentto In andthusisnonsingular.
(c)Ifwetransform E3 toreducedrowechelonform,weobtain In .Hence E3 isrowequivalentto In andthusisnonsingular.
8. ⎡ ⎢ ⎢ ⎣ 1 aa2 a3 01 aa2 001 a 0001
10.(a) ⎡ ⎣
12. s =0, ±√2.
13.Foranyangle θ ,cos θ andsin θ areneversimultaneouslyzero.Thusatleastoneelementincolumn1 isnotzero.Assumecos θ =0.(Ifcos θ =0,theninterchangerows1and2andproceedinasimilar mannertothatdescribedbelow.)Toshowthatthematrixisnonsingularanddetermineitsinverse, weput
intoreducedrowechelonform.Applyrowoperations 1 cos θ timesrow1andsin θ timesrow1addedto row2toobtain
Since
the(2, 2)-elementisnotzero.Applyingrowoperationscos θ timesrow2and sin θ cos θ timesrow2 addedtorow1weobtain
Itfollowsthatthematrixisnonsingularanditsinverseis
14.(a) A(u + v )= Au + Av = 0 + 0 = 0
(b) A(u v )= Au Av = 0 0 = 0.
(c) A(r u)= r (Au)= r 0 = 0.
(d) A(r u + sv )= r (Au)+ s(Av )= r 0 + s0 = 0
15.If Au = b and Av = b,then A(u v )= Au Av = b b =
16.Supposeatsomepointintheprocessofreducingtheaugmentedmatrixtoreducedrowechelonform weencounterarowwhosefirst n entriesarezerobutwhose(n +1)stentryissomenumber c =0.The correspondinglinearequationis 0 · x1 + ··· +0 · xn = c or0= c.
Thisequationhasnosolution,thusthelinearsystemisinconsistent.
17.Let u beonesolutionto Ax = b.Since A issingular,thehomogeneoussystem Ax = 0 hasanontrivial solution u0 .Thenforanyrealnumber r , v = r u0 isalsoasolutiontothehomogeneoussystem.Finally, byExercise29,Sec.2.2,foreachoftheinfinitelymanyvectors v ,thevector w = u + v isasolution tothenonhomogeneoussystem Ax = b.
18. s =1, t =1.
20.Ifanyofthediagonalentriesof L or U iszero,therewillnotbeauniquesolution.
21.Theouterproductof X and Y canbewrittenintheform
Ifeither X = O or Y = O ,then XY T = O .Thusassumethatthereisatleastonenonzerocomponent in X ,say xi ,andatleastonenonzerocomponentin Y ,say yj .Then 1 xi Row i (XY T )makesthe ith rowexactly Y T .Sincealltheotherrowsaremultiplesof Y T ,rowoperationsoftheform xk Ri + Rp , for p = i,canbeperformedtozeroouteverythingbutthe ithrow.Itfollowsthateither XY T isrow equivalentto O ortoamatrixwith n 1zerorows.
ChapterReviewforChapter2,p.138
TrueorFalse
1.False.2.True.3.False.4.True.5.True. 6.True.7.True.8.True.9.True.10.False.
Quiz
1. ⎡ ⎣ 102 013 000 ⎤ ⎦
2.(a)No. (b)Infinitelymany. (c)No. (d) x = ⎡ ⎢ ⎢ ⎣ 6+2r +7s r 3s s ⎤ ⎥ ⎥ ⎦ ,where r and s areanyrealnumbers.
3. k =6.
7.Possibleanswers:Diagonal,zero,orsymmetric.