LGEBRA
WITH A PPLICATIONS
TENTH EDITION
Steven J. Leon
University of Massachusetts Dartmouth
Lisette G. de Pillis
Harvey Mudd College
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ISBN-13: 978-0-13-526050-0
ISBN-10: 0-13-526050-7
Preface
Thissolutionsmanualisdesignedtoaccompanythetentheditionof LinearAlgebrawithApplications byStevenJ.LeonandLisettedePillis.Themanualcontainsthecompletesolutionstoallofthe nonroutineexercisesandChaptertestquestionsinthefirstsevenchaptersthebook.Eachofthose chaptersalsoincludesasetofMATLABcomputerexercises.MostoftheMATLABcomputations arestraightforward.andconsequentlythecomputationalresultsarenotincludedinthismanual. However,theMATLABExercisesalsoincludequestionsrelatedtothecomputations.Thepurpose ofthequestionsistoemphasizethesignificanceofthecomputations.Thismanualdoesprovidethe answerstomostofthesequestions.
Matricesand
(d)4x1 3x2 + x3 +2x4 =4 3x1 + x2 5x3 +6x4 =5 x1 + x2 +2x3 +4x4 =8 5x1 + x2 +3x3 2x4 =7
9. Giventhesystem
onecaneliminatethevariable x2 bysubtractingthefirstrowfromthesecond.Onethen obtainstheequivalentsystem
1x1 + x2 = b1 (m1 m2)x1 = b2 b1
(a)If m1 = m2,thenonecansolvethesecondequationfor x1 x1 = b2 b1 m1 m2
Onecanthenplugthisvalueof x1 intothefirstequationandsolvefor x2.Thus,if m1 = m2,therewillbeauniqueorderedpair(x1,x2)thatsatisfiesthetwoequations.
(b)If m1 = m2,thenthe x1 termdropsoutinthesecondequation 0= b2 b1
Thisispossibleifandonlyif b1 = b2.
(c)If m1 = m2,thenthetwoequationsrepresentlinesintheplanewithdifferentslopes. Twononparallellinesintersectinapoint.Thatpointwillbetheuniquesolutionto thesystem.If m1 = m2 and b1 = b2,thenbothequationsrepresentthesamelineand consequentlyeverypointonthatlinewillsatisfybothequations.If m1 = m2 and b1 = b2, thentheequationsrepresentparallellines.Sinceparallellinesdonotintersect,thereis nopointonbothlinesandhencenosolutiontothesystem.
10. Thesystemmustbeconsistentsince(0, 0)isasolution.
11. Alinearequationin3unknownsrepresentsaplaneinthreespace.Thesolutionsettoa3 × 3 linearsystemwouldbethesetofallpointsthatlieonallthreeplanes.Iftheplanesare paralleloroneplaneisparalleltothelineofintersectionoftheothertwo,thenthesolution setwillbeempty.Thethreeequationscouldrepresentthesameplaneorthethreeplanes couldallintersectinaline.Ineithercasethesolutionsetwillcontaininfinitelymanypoints. Ifthethreeplanesintersectinapoint,thenthesolutionsetwillcontainonlythatpoint.
2 ROWECHELONFORM
2. (b)Thesystemisconsistentwithauniquesolution(4, 1).
4. (b) x1 and x3 areleadvariablesand x2 isafreevariable. (d) x1 and x3 areleadvariablesand x2 and x4 arefreevariables. (f) x2 and x3 areleadvariablesand x1 isafreevariable.
5. (l)Thesolutionis(0, 1 5, 3 5).
6. (c)Thesolutionsetconsistsofallorderedtriplesoftheform(0, α,α).
7. Ahomogeneouslinearequationin3unknownscorrespondstoaplanethatpassesthrough theoriginin3-space.Twosuchequationswouldcorrespondtotwoplanesthroughtheorigin. Ifoneequationisamultipleoftheother,thenbothrepresentthesameplanethroughthe originandeverypointonthatplanewillbeasolutiontothesystem.Ifoneequationisnot amultipleoftheother,thenwehavetwodistinctplanesthatintersectinalinethroughthe
origin.Everypointonthelineofintersectionwillbeasolutiontothelinearsystem.Soin eithercasethesystemmusthaveinfinitelymanysolutions.
Inthecaseofanonhomogeneous2 × 3linearsystem,theequationscorrespondtoplanes thatdonotbothpassthroughtheorigin.Ifoneequationisamultipleoftheother,thenboth representthesameplaneandthereareinfinitelymanysolutions.Iftheequationsrepresent planesthatareparallel,thentheydonotintersectandhencethesystemwillnothaveany solutions.Iftheequationsrepresentdistinctplanesthatarenotparallel,thentheymust intersectinalineandhencetherewillbeinfinitelymanysolutions.Sotheonlypossibilities foranonhomogeneous2 × 3linearsystemare0orinfinitelymanysolutions.
9. (a)Sincethesystemishomogeneousitmustbeconsistent.
13. Ahomogeneoussystemisalwaysconsistentsinceithasthetrivialsolution(0,..., 0).Ifthe reducedrowechelonformofthecoefficientmatrixinvolvesfreevariables,thentherewillbe infinitelymanysolutions.Iftherearenofreevariables,thenthetrivialsolutionwillbethe onlysolution.
14. Anonhomogeneoussystemcouldbeinconsistentinwhichcasetherewouldbenosolutions. Ifthesystemisconsistentandunderdetermined,thentherewillbefreevariablesandthis wouldimplythatwewillhaveinfinitelymanysolutions.
16. Ateachintersection,thenumberofvehiclesenteringmustequalthenumberofvehiclesleaving inorderforthetraffictoflow.Thisconditionleadstothefollowingsystemofequations
Ifweaddallfourequations,weget
andhence
17. If(c1,c2)isasolution,then
Multiplyingbothequationsthroughby α,oneobtains
Thus(αc1,αc2)isalsoasolution.
18. (a)If x4 =0,then x1, x2,and x3 willallbe0.Thusifnoglucoseisproduced,thenthere isnoreaction.(0, 0, 0, 0)isthetrivialsolutioninthesensethatiftherearenomoleculesof carbondioxideandwater,thentherewillbenoreaction.
(b)Ifwechooseanothervalueof x4,say x4 =2,thenweendupwithsolution x1 =12, x2 =12, x3 =12, x4 =2.Notetheratiosarestill6:6:6:1.
3 MATRIXARITHMETIC
9. (b) x =(2, 1)T isasolutionsince b =2a1 + a2.Therearenoothersolutionssincetheechelon formof A isstrictlytriangular.
(c)Thesolutionto Ax = c is x =( 5 2 , 1 4 )T .Therefore c = 5 2 a1 1 4 a2
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11. Thegiveninformationimpliesthat
arebothsolutionstothesystem.Sothesystemisconsistentandsincethereismorethanone solution,therowechelonformof A mustinvolveafreevariable.Aconsistentsystemwitha freevariablehasinfinitelymanysolutions.
12. Thesystemisconsistentsince x =(1, 1, 1, 1)T isasolution.Thesystemcanhaveatmost3 leadvariablessince A onlyhas3rows.Therefore,theremustbeatleastonefreevariable.A consistentsystemwithafreevariablehasinfinitelymanysolutions.
13. (a)Itfollowsfromthereducedrowechelonformthatthefreevariablesare x2, x4, x5.Ifwe set x2 = a, x4 = b, x5 = c,then x1 =
andhencethesolutionconsistsofallvectorsoftheform x =( 2 2a 3b c,a, 5 2b 4c,b,c)T
(b)Ifwesetthefreevariablesequalto0,then x0 =( 2,
,
,
, 0)T isasolutionto Ax = b andhence
14. If w3 istheweightgiventoprofessionalactivities,thentheweightsforresearchandteaching shouldbe w1 =3w3 and w2 =2w3.Notethat
sotheweightgiventoresearchis1.5timestheweightgiventoteaching.Sincetheweights mustalladdupto1,wehave
andhenceitfollowsthat w3 =
.If C isthematrixintheexampleproblem fromtheAnalyticHierarchyProcessApplication,thentheratingvector r iscomputedby multiplying C timestheweightvector w.
15. AT isan n × m matrix.Since AT has m columnsand A has m rows,themultiplication AT A ispossible.Themultiplication AAT ispossiblesince A has n columnsand AT has n rows.
16. If A isskew-symmetric,then AT = A.Sincethe(j,j)entryof AT is ajj andthe(j,j)entry of A is ajj ,itfollowsthat ajj = ajj foreach j andhencethediagonalentriesof A must allbe0.
17. Thesearchvectoris x =(1, 0, 1, 0, 1, 0)T .Thesearchresultisgivenbythevector y = AT x =(1, 2, 2, 1, 1, 2, 1)T
The ithentryof y isequaltothenumberofsearchwordsinthetitleofthe ithbook.
18. If α = a21/a11,then
Theproductwillequal A provided
Thuswemustchoose
4 MATRIXALGEBRA
1. (a)(A + B)2 =(A + B)(A + B)=(A + B)A +(A + B)B = A2 + BA + AB + B2
Forrealnumbers, ab + ba =2ab;however,withmatrices AB + BA isgenerallynotequal to2AB. (b)
Forrealnumbers, ab ba =0;however,withmatrices AB BA isgenerallynotequal to O.
2. Ifwereplace a by A and b bytheidentitymatrix, I,thenbothruleswillwork,since (A + I)2 = A2 + IA + AI + B2 = A2 + AI + AI +
and
3. Therearemanypossiblechoicesfor A and B.Forexample,onecouldchoose
Moregenerallyif
then AB = O foranychoiceofthescalars a, b, c, d, e
4. Toconstructnonzeromatrices A, B, C withthedesiredproperties,firstfindnonzeromatrices C and D suchthat DC = O (seeExercise3).Next,foranynonzeromatrix A,set B = A + D Itfollowsthat
5. A2 × 2symmetricmatrixisoneoftheform
Chapter1 • MatricesandSystemsofEquations
Thus
If A2 = O,thenitsdiagonalentriesmustbe0.
Thus a = b = c =0andhence A = O
6. Let
Itfollowsthat
Ifweset
thenitfollowsthat
b11c11 + b12c21 b11c12 + b12c22 b21c11 + b22c21 b21c12 + b22c22
e11 = a11(b11c11 + b12c21)+ a12(
andhence
10. (a)Thematrix C issymmetricsince
(b)Thematrix D issymmetricsince
(c)Thematrix E = AB isnotsymmetricsince
andingeneral, AB = BA
(d)Thematrix F issymmetricsince
(e)Thematrix G issymmetricsince
(f)Thematrix H isnotsymmetricsince
11. (a)Thematrix A issymmetricsince
(b)Thematrix B isnotsymmetricsince
(c)Thematrix D issymmetricsince
(d)Thematrix E issymmetricsince
(e)Thematrix F issymmetricsince
(e)Thematrix G isnotsymmetric.
13. (b)
35 2 3
14. If A werenonsingularand AB = A,thenitwouldfollowthat A 1AB = A 1A andhence that B = I.Soif B = I,then A mustbesingular.
15. Since A 1A = AA 1 = I
itfollowsfromthedefinitionthat A 1 isnonsingularanditsinverseis A
16. Since
itfollowsthat
17. If Ax = Ay and x = y,then A mustbesingular,forif A werenonsingular,thenwecould multiplyby A 1 andget A 1Ax = A 1
18. For m =1,
Assumetheresultholdsinthecase m = k,thatis, (Ak) 1 =(A 1)k
Itfollowsthat
and
andtheresultfollowsbymathematicalinduction.
19. If A2 = O,then
and
I A isnonsingularand(
20. If Ak+1 = O,then
Therefore I A isnonsingularand(I A) 1 = I + A + A2 + + Ak 21. Since
and
itfollowsthat R isnonsingularand R 1 = RT 22.
23.
24. Ineachcase,ifyousquarethegivenmatrix,youwillendupwiththesamematrix.
25. (a)If A2 = A,then
(b)If A2 = A,then
26. (a)
and
Sinceeachdiagonalentryof D isequaltoeither0or1,itfollowsthat d2 jj = djj ,for j =1,...,n andhence D2 = D
(b)If A = XDX 1,then A2 =(XDX 1)(XDX 1)= XD(X 1X)DX 1 = XDX 1 = A Copyright ©2020 Pearson Education, Inc.
27. If A isaninvolution,then A2 = I anditfollowsthat B2 = 1 4 (I + A)2 = 1 4 (I +2A + A2)= 1 4 (2I +2A)= 1 2 (I + A)= B
So B and C arebothidempotent.
BC = 1 4 (I + A)(I A)= 1 4 (I +
28. (AT A)T = AT (AT )T = AT A
(AAT )T =(AT )T AT = AAT
29. Let A and B besymmetric n × n matrices.If(AB)T = AB,then
BA = BT AT =(AB)T = AB
Conversely,if BA = AB,then (AB)T = BT AT = BA = AB
30. (a) BT =(A + AT )T = AT +(AT )T = AT + A = B CT =(
(b) A = 1 2 (A + AT )+ 1 2 (A AT )
34. False.Forexample,if
however, A = B.
35. False.Forexample,if
thenitiseasytoseethatboth A and B mustbesingular,however, A + B = I,whichis nonsingular.
36. True.If A and B arenonsingular,thentheirproduct AB mustalsobenonsingular.Usingthe resultfromExercise23,wehavethat(AB)T isnonsingularand((AB)T ) 1 =((AB) 1)T .It followsthenthat
5 ELEMENTARYMATRICES
2. (a) 0 1 10 ,typeI Copyright ©2020 Pearson Education, Inc.
(b)Thegivenmatrixisnotanelementarymatrix.Itsinverseisgivenby
5. (c)Since
where F and E areelementarymatrices,itfollowsthat C isrowequivalentto A
7. A canbereducedtotheidentitymatrixusingthreerowoperations
Theelementarymatricescorrespondingtothethreerowoperationsare
andhence
(d) XA + C = X
13. (a)If E isanelementarymatrixoftypeIortypeII,then E issymmetric.Thus ET = E is anelementarymatrixofthesametype.If E istheelementarymatrixoftypeIIIformed byadding α timesthe ithrowoftheidentitymatrixtothe jthrow,then ET isthe elementarymatrixoftypeIIIformedfromtheidentitymatrixbyadding α timesthe jth rowtothe ithrow.
(b)Ingeneral,theproductoftwoelementarymatriceswillnotbeanelementarymatrix. Generally,theproductoftwoelementarymatriceswillbeamatrixformedfromthe identitymatrixbytheperformanceoftworowoperations.Forexample,if
then E1 and E2 areelementarymatrices,but
isnotanelementarymatrix.
14. If T = UR,then
Since U and R areuppertriangular
If i>j,then
Therefore T isuppertriangular.
If i = j,then
15. Ifweset x =(2, 1 4)T ,then
Thus x isanonzerosolutiontothesystem Ax = 0.Butifahomogeneoussystemhasa nonzerosolution,thenitmusthaveinfinitelymanysolutions.Inparticular,if c isanyscalar, then cx isalsoasolutiontothesystemsince
Since Ax = 0 and x = 0,itfollowsthatthematrix A mustbesingular.(SeeTheorem1.5.2)
16. If a1 =3a2 2a3,then
Therefore x =(1, 3, 2)T isanontrivialsolutionto Ax = 0.ItfollowsfromTheorem1.5.2 that A mustbesingular.
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17. If x0 = 0 and Ax0 = Bx0,then Cx0 = 0 anditfollowsfromTheorem1.5.2that C mustbe singular.
18. If B issingular,thenitfollowsfromTheorem1.5.2thatthereexistsanonzerovector x such that Bx = 0.If C = AB,then Cx = ABx = A0 = 0
Thus,byTheorem1.5.2, C mustalsobesingular.
19. (a)If U isuppertriangularwithnonzerodiagonalentries,thenusingrowoperationII, U can betransformedintoanuppertriangularmatrixwith1’sonthediagonal.Rowoperation IIIcanthenbeusedtoeliminatealloftheentriesabovethediagonal.Thus, U isrow equivalentto I andhenceisnonsingular.
(b)Thesamerowoperationsthatwereusedtoreduce U totheidentitymatrixwilltransform I into U 1.RowoperationIIappliedto I willjustchangethevaluesofthediagonal entries.WhentherowoperationIIIstepsreferredtoinpart(a)areappliedtoadiagonal matrix,theentriesabovethediagonalarefilledin.Theresultingmatrix, U 1,willbe uppertriangular.
20. Since A isnonsingularitisrowequivalentto I.Hence,thereexistelementarymatrices E1,E2,...,Ek suchthat
Itfollowsthat
and
Thesamerowoperationsthatreduce A to I,willtransform B to C.Therefore,thereduced rowechelonformof(A | B)willbe(I | C).
21. (a)Ifthediagonalentriesof D1 are α1,α2,...,αn andthediagonalentriesof D2 are β1,β2,...,βn,then D1D2 willbeadiagonalmatrixwithdiagonalentries α1β1,...,αnβn and D2D1 willbeadiagonalmatrixwithdiagonalentries β1α1,β2α2,...,βnαn.Since thetwohavethesamediagonalentries,itfollowsthat D1D2 = D2D1 (b)
AB = A(a0I + a1A + + akAk) = a0A + a1
= BA
22. If A issymmetricandnonsingular,then (A 1)T =(A 1)T (AA 1)=((A 1)
AT )A 1 = A 1
23. If A isrowequivalentto B,thenthereexistelementarymatrices E1,E2,...,Ek suchthat
Eachofthe Ei’sisinvertibleand E 1 i isalsoanelementarymatrix(Theorem1.4.1).Thus
andhence B isrowequivalentto A
24. (a)If A isrowequivalentto B,thenthereexistelementarymatrices E1,E2,...,Ek such that
Since B isrowequivalentto C,thereexistelementarymatrices H1,H2,...,Hj suchthat B = Hj Hj 1 H1C
Thus
= EkEk 1 E1Hj Hj 1 H1C
andhence A isrowequivalentto C
(b)If A and B arenonsingular n × n matrices,then A and B arerowequivalentto I.Since A isrowequivalentto I and I isrowequivalentto B,itfollowsfrompart(a)that A is rowequivalentto B.
25. If U isanyrowechelonformof A,then A canbereducedto U usingrowoperations,so A isrowequivalentto U .If B isrowequivalentto A,thenitfollowsfromtheresultin Exercise24(a)that B isrowequivalentto U
26. If B isrowequivalentto A,thenthereexistelementarymatrices E1,E2,...,Ek suchthat
Let M = EkEk 1 E1.Thematrix M isnonsingularsinceeachofthe Ei’sisnonsingular. Conversely,supposethereexistsanonsingularmatrix M suchthat B = MA.Since M isnonsingular,itisrowequivalentto I.Thus,thereexistelementarymatrices E1,E2,...,Ek suchthat
M = EkE
Itfollowsthat
Therefore, B isrowequivalentto A
27. If A isnonsingular,then A isrowequivalentto I.If B isrowequivalentto A,thenusing theresultfromExercise24(a),wecanconcludethat B isrowequivalentto I.Therefore, B mustbenonsingular.Soitisnotpossiblefor B tobesingularandalsoberowequivalentto anonsingularmatrix.
28. (a)Thesystem V c = y isgivenby
Comparingthe ithrowofeachside,wehave
c1 + c2xi + + cn+1xn i = yi Thus
p(xi)= yi i =1, 2,...,n +1
(b)If x1,x2,...,xn+1 aredistinctand V c = 0,thenwecanapplypart(a)with y = 0.Thus if p(x)= c1 + c2x + ··· + cn+1xn,then
p(xi)=0 i =1, 2,...,n +1
Thepolynomial p(x)has n +1roots.Sincethedegreeof p(x)islessthan n +1, p(x) mustbethezeropolynomial.Hence
c1 = c2 = = cn+1 =0
Sincethesystem V c = 0 hasonlythetrivialsolution,thematrix V mustbenonsingular.
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29. True.If A isrowequivalentto I,then A isnonsingular,soif AB = AC,thenwecanmultiply bothsidesofthisequationby A 1 .
1AB = A 1AC
= C
30. True.If E and F areelementarymatrices,thentheyarebothnonsingularandtheproduct oftwononsingularmatricesisanonsingularmatrix.Indeed, G 1 = F 1E 1
31. True.If a + a2 = a3 +2a4,then
Ifwelet x =(1, 1, 1, 2)T ,then x isasolutionto Ax = 0.Since x = 0 thematrix A must besingular.
32. False.Let I bethe2 × 2identitymatrixandlet A = I, B = I,and
Since B and C arenonsingular,theyarebothrowequivalentto A;however,
issingular,soitcannotberowequivalentto A
6 PARTITIONEDMATRICES
Theblockmultiplicationisperformedasfollows:
(b)Since y
=1, 2, 3,theouterproductexpansionof YX T isjustthe transposeoftheouterproductexpansionof XY T .Thus
7. Itispossibletoperformbothblockmultiplications.Toseethis,suppose A11 isa k ×r matrix, A12 isa k × (n r)matrix, A21 isan(m k) × r matrixand A22 is(m k) × (n r).Itis possibletoperformtheblockmultiplicationof AAT sincethematrixmultiplications A11AT 11, A11AT 21, A12AT 12, A12
,
22 areallpossible.Itispossibleto performtheblockmultiplicationof AT A sincethematrixmultiplications AT 11A11, AT 11A12, AT 21A21, AT 21A11, AT 12A12, AT 22A21, AT 22A22 areallpossible.
8. AX = A(x1, x2,..., xr)=(Ax1,Ax2,...,Axr)
B =(b1, b2,..., br)
AX = B ifandonlyifthecolumnvectorsof AX and B areequal Axj = bj j =1,...,r
9. (a)Since D isadiagonalmatrix,its jthcolumnwillhave djj inthe jthrowandtheother entrieswillallbe0.Thus dj = djj ej for j =1,...,n (b)
= A(d11e1,d22e2,...,dnn
10. (a)
(b)Ifwelet X = U Σ,then
anditfollowsthat
If
then
Let
Since AB = BA = I,itfollowsthat B = A 1 .
12. Let 0 denotethezerovectorin Rn.If A issingular,thenthereexistsavector x1 = 0 such that Ax1 = 0.Ifweset
then
ByTheorem1.5.2, M mustbesingular.Similarly,if B issingular,thenthereexistsavector x2 = 0 suchthat Bx2 = 0.Soifweset
then x isanonzerovectorand M x isequaltothezerovector. 15.
andhence
16. Theblockformof S 1 isgivenby
Itfollowsthat
17. Theblockmultiplicationofthetwofactorsyields
Ifweequatethismatrixwiththeblockformof A andsolvefor B and C,weget
Tocheckthatthisworksnotethat
andhence
18. Inorderfortheblockmultiplicationtowork,wemusthave
= S and YM = T
Sinceboth B and M arenonsingular,wecansatisfytheseconditionsbychoosing X = SB 1 and Y = TM 1
19. (a)
(c)Itfollowsfromparts(a)and(b)that
20. If Ax = 0 forall x ∈ Rn,then
aj = Aej = 0 for j =1,...,n andhence A mustbethezeromatrix.
21. If Bx = Cx forall x ∈ Rn then (B C)x =
ItfollowsfromExercise20that
22. (a)
then
and
MATLABEXERCISES
1. Inparts(a),(b),(c)itshouldturnoutthat A1= A4and A2= A3.Inpart(d) A1= A3 and A2= A4.Exactequalitywillnotoccurinparts(c)and(d)becauseofroundofferror.
2. Thesolution x obtainedusingthe \ operationwillbemoreaccurateandyieldthesmaller residualvector.Thecomputationof x isalsomoreefficientsincethesolutioniscomputed usingGaussianeliminationwithpartialpivotingandthisinvolveslessarithmeticthancomputingtheinversematrixandmultiplyingittimes b
3. (a)Since Ax = 0 and x = 0,itfollowsfromTheorem1.5.2that A issingular.
(b)Thecolumnsof B areallmultiplesof x.Indeed,
andhence
(c)If D = B + C,then
AD = AB + AC = O + AC = AC
4. Byconstruction, B isuppertriangularwhosediagonalentriesareallequalto1.Thus B is rowequivalentto I andhence B isnonsingular.Ifonechanges B bysetting b10,1 = 1/256 andcomputes Bx,theresultisthezerovector.Since x = 0,thematrix B mustbesingular.
5. (a)Since A isnonsingular,itsreducedrowechelonformis I.If E1,...,Ek areelementary matricessuchthat Ek E1A = I,thenthesesamematricescanbeusedtotransform (A b)toitsreducedrowechelonform U .Itfollowsthenthat
U = Ek E1(A b)= A 1(A b)=(IA 1b)
Thus,thelastcolumnof U shouldbeequaltothesolution x ofthesystem Ax = b
(b)Afterthethirdcolumnof A ischanged,thenewmatrix A isnowsingular.Examining thelastrowofthereducedrowechelonformoftheaugmentedmatrix(A b),weseethat thesystemisinconsistent.
(c)Thesystem Ax = c isconsistentsince y isasolution.Thereisafreevariable x3,sothe systemwillhaveinfinitelymanysolutions.
(f)Thevector v isasolutionsince
v = A(w +3z)= Aw +3Az = c
Forthissolution,thefreevariable x3 = v3 =3.Todeterminethegeneralsolutionjust set x = w + tz.Thiswillgivethesolutioncorrespondingto x3 = t foranyrealnumber t.
6. (c)Therewillbenowalksofevenlengthfrom Vi to Vj whenever i + j isodd.
(d)Therewillbenowalksoflength k from Vi to Vj whenever i + j + k isodd.
(e)Theconjectureisstillvalidforthegraphcontainingtheadditionaledges.
(f)Iftheedge {V6,V8} isincluded,thentheconjectureisnolongervalid.Thereisnowa walkoflength1from V6 to V8 and i + j + k =6+8+1isodd.
8. Thechangeinpart(b)shouldnothaveasignificanteffectonthesurvivalpotentialforthe turtles.Thechangeinpart(c)willeffectthe(2, 2)and(3, 2)oftheLesliematrix.Thenew valuesfortheseentrieswillbe l22 =0.9540and l32 =0.0101.Withthesevalues,theLeslie populationmodelshouldpredictthatthesurvivalperiodwilldoublebuttheturtleswillstill eventuallydieout.
9. (b) x1 = c V x2
10. (b)
Thiscanbeprovedusingmathematicalinduction.Inthecase k =1
Iftheresultholdsfor
Chapter1 • MatricesandSystemsofEquations
Itfollowsbymathematicalinductionthattheresultholdsforallpositiveintegers k. (b)
11. (a)Byconstruction,theentriesof A wereroundedtothenearestinteger.Thematrix B = ATA mustalsohaveintegerentriesanditissymmetricsince
(b)
Itfollowsthat
CHAPTERTESTA
1. Thestatementisfalse.Iftherowechelonformhasfreevariablesandthelinearsystem isconsistent,thentherewillbeinfinitelymanysolutions.However,itispossibletohavean inconsistentsystemwhosecoefficientmatrixwillreducetoanechelonformwithfreevariables. Forexample,if
then A involvesonefreevariable,butthesystem Ax = b isinconsistent.
2. Thestatementistruesincethezerovectorwillalwaysbeasolution.
3. Thestatementistrue.Amatrix A isnonsingularifandonlyifitisrowequivalenttothe I (theidentitymatrix). A willberowequivalentto I ifandonlyifitsreducedrowechelon formis I
4. Thestatementistrue.If A isnonsingular,then A isrowequivalentto I.Sothereexist elementarymatrices E1,E2,...,Ek,suchthat
5. Thestatementisfalse.Forexample,if A = I and B = I,thematrices A and B areboth nonsingular,but A + B = O issingular.
6. Thestatementisfalse.Forexample,if A isanymatrixoftheform
Then A = A 1 .
7. Thestatementisfalse. (A B)2 = A2 BA AB + B2 = A2 2AB + B2 sinceingeneral BA = AB.Forexample,if
8. Thestatementisfalse.If A isnonsingularand AB = AC,thenwecanmultiplybothsidesof theequationby A 1 andconcludethat B = C.However,if A issingular,thenitispossible tohave AB = AC and B = C.Forexample,if
9. Thestatementisfalse.Ingeneral, AB and BA areusuallynotequal,soitispossiblefor AB = O and BA tobeanonzeromatrix.Forexample,if
then
10. Thestatementistrue.If x =(1, 2, 1)T ,then x = 0 and Ax = 0,so A mustbesingular.
11. Thestatementistrue.If b = a1 + a3 and x =(1, 0, 1)T ,then
So x isasolutionto Ax = b
12. Thestatementistrue.If b = a1 + a2 +
3,then x =(1, 1, 1)T isasolutionto Ax = b,since
If a2 = a3,thenwecanalsoexpress b asalinearcombination
Thus y =(1, 0, 2)T isalsoasolutiontothesystem.However,ifthereismorethanone solution,thenthereducedrowechelonformof A mustinvolveafreevariable.Aconsistent systemwithafreevariablemusthaveinfinitelymanysolutions.
13. Thestatementistrue.Anelementarymatrix E oftypeIortypeIIissymmetric.Soineither casewehave ET = E iselementary.If E isanelementarymatrixoftypeIIIformedfrom theidentitymatrixbyaddinganonzeromultiple c ofrow k torow j,then ET willbethe elementarymatrixoftypeIIIformedfromtheidentitymatrixbyadding c timesrow j to row k
14. Thestatementisfalse.Anelementarymatrixisamatrixthatisconstructedbyperforming exactlyoneelementaryrowoperationontheidentitymatrix.Theproductoftwoelementary matriceswillbeamatrixformedbyperforming two elementaryrowoperationsontheidentity matrix.Forexample,
areelementarymatrices,however;
isnotanelementarymatrix.
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15. Thestatementistrue.Therowvectorsof A are x1yT ,x2yT ,...,xnyT .Note,alloftherow vectorsaremultiplesof yT .Since x and y arenonzerovectors,atleastoneoftheserow vectorsmustbenonzero.However,ifanynonzerorowispickedasapivotrow,thensinceall oftheotherrowsaremultiplesofthepivotrow,theywillallbeeliminatedinthefirststep ofthereductionprocess.Theresultingrowechelonformwillhaveexactlyonenonzerorow.
CHAPTERTESTB
Thefreevariablesare x
andhencethesolutionsetconsistsofallvectorsoftheform
2. (a)Alinearequationin3unknownscorrespondstoaplanein3-space.
(b)Given2equationsin3unknowns,eachequationcorrespondstoaplane.Ifoneequation isamultipleoftheother,thentheequationsrepresentthesameplaneandanypointon thethatplanewillbeasolutiontothesystem.Ifthetwoplanesaredistinct,thenthey areeitherparallelortheyintersectinaline.Iftheyareparalleltheydonotintersect,so thesystemwillhavenosolutions.Iftheyintersectinaline,thentherewillbeinfinitely manysolutions.
(c)Ahomogeneouslinearsystemisalwaysconsistentsinceithasthetrivialsolution x = 0 Itfollowsfrompart(b)thenthatahomogeneoussystemof2equationsin3unknowns musthaveinfinitelymanysolutions.Geometricallythe2equationsrepresentplanesthat bothpassthroughtheorigin,soiftheplanesaredistincttheymustintersectinaline.
3. (a)Ifthesystemisconsistentandtherearetwodistinctsolutions,thentheremustbeafree variableandhencetheremustbeinfinitelymanysolutions.Infact,allvectorsofthe form x = x1 + c(x1 x2)willbesolutionssince Ax = Ax1 + c(Ax1 Ax2)= b + c(b b)= b
(b)Ifweset z = x1 x2,then z = 0 and Az = 0.Therefore,itfollowsfromTheorem1.5.2 that A mustbesingular.
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4. (a)Thesystemwillbeconsistentifandonlyifthevector b =(3, 1)T canbewrittenasa linearcombinationofthecolumnvectorsof A.Linearcombinationsofthecolumnvectors of A arevectorsoftheform
Since b isnotamultipleof(1, 2)T thesystemmustbeinconsistent.
(b)Toobtainaconsistentsystem,choose b tobeamultipleof(1, 2)T .Ifthisisdonethe secondrowoftheaugmentedmatrixwillzerooutintheeliminationprocessandyouwill endupwithoneequationin2unknowns.Thereducedsystemwillhaveinfinitelymany solutions.
5. (a)Totransform A to B,youneedtointerchangethesecondandthirdrowsof A.The elementarymatrixthatdoesthisis
(b)Totransform A to C usingacolumnoperation,youneedtosubtracttwicethesecond columnof A fromthefirstcolumn.Theelementarymatrixthatdoesthisis
6. If b =3a1 + a2 +4a3,then b isalinearcombinationofthecolumnvectorsof A anditfollows fromtheconsistencytheoremthatthesystem Ax = b isconsistent.Infact, x =(3, 1, 4)T is asolutiontothesystem.
7. If a1 3a2 +2a3 = 0,then x =(1, 3, 2)T isasolutionto Ax = 0.ItfollowsfromTheorem1.5.2that A mustbesingular.
8. If
9. Ingeneral,theproductoftwosymmetricmatricesisnotnecessarilysymmetric.Forexample, if
then A and B arebothsymmetricbuttheirproduct
isnotsymmetric.
10. If E and F areelementarymatrices,thentheyarebothnonsingularandtheirinversesare elementarymatricesofthesametype.If C = EF ,then C isaproductofnonsingularmatrices, so C isnonsingularand C 1 = F 1E 1 11.
12. (a)Thecolumnpartitionof A andtherowpartitionof B mustmatchup,so k mustbeequal to5.Thereisreallynorestrictionon r,itcanbeanyintegerintherange1 ≤ r ≤ 9.In fact, r =10willworkwhen B hasblockstructure
(b)The(2,2)blockoftheproductisgivenby
Determinants
1 THEDETERMINANTOFAMATRIX
1. (c)det(A)= 3
7. Giventhat a11 =0and a21 =0,letusinterchangethefirsttworowsof A andalsomultiply thethirdrowthroughby a21.Weendupwiththematrix
Nowifweadd a31 timesthefirstrowtothethird,weobtainthematrix
Thismatrixwillberowequivalentto I ifandonlyif
Thustheoriginalmatrix A willberowequivalentto I ifandonlyif