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# 1st - CHAPTER 2 Systems of Linear Equations EKEHGISESETZJI...

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Unformatted text preview: CHAPTER 2 Systems of Linear Equations EKEHGISESETZJI 1D. (3.} z=3wlﬂt,y=t1whem-ooﬁlcoo. {h}:1=3-3a+12:t,mg=s,:c3=t,whm—oo{s.t{oo. [o] :1=r132:5,:tg=t.:4-2ﬂ-—4r-23-3t.whoro-oo<:r,a,t{oo. {d} U=h-W=33E=—¢1-tn+5ia-7h.y=latz=¢m‘ﬂ'hﬁm ~W€3ht2l¢ag hqoo. 26. {a} BisobtainedﬁomAbyintorohnngingtheﬁmtmdthlrdmm. Aiauhtninodﬁ'omﬂhy interChHIg’ingthEﬁmtnndthirdm. {b} BhobtﬁnedﬁmnAbymﬁﬁplyingthnthirdmwhyﬁ.Aiaohtninodh'omﬂhymulﬂplytng thnthirdmwbjr i 31.91] 3cg+oz+2cg— {24—5 Dz+3o3+2ci=ﬂ -C1+ﬂ2 +5¢4=5 2c1+¢2+263 =5 DB. [21} 'Ir-no.chorenrenzﬂcolutnnsﬁhenthoﬁmtna—loolumnnoononpondtothooooﬁoientn olthnmriobloothatappenrinthooquntlonsondthohotoolumnoormpondntothemmtm that appenronthoright-handsidonlthnmualaign. {b} Mao. Hofmhgtoﬂxmnple 6: Thomunnoooflinnnrayotom appearing in tho [Eli-hand column nl] ham: the some solution not, but the wounding augmented matrioen appearing in the right-hand column are all dlﬂemnt. {4:} Palm. Moltiphringnrnw of the augmntod matrixbyoerommoponm tomnltiplying both aid-esoftho corresponding oquntionbgr men. But thlniaequivnlent tn discardingonn of the equations! (d) Thin.chearstmnismnoiotont,onnoansolvefoctwnofthominblnaintormofthathirtl orﬂfl‘nrtherrodnndmcyinprmnt]farunaol‘thmvnrinhluintermsofthoothertm. loony om.thernisatlnastnnn “Eros"vnﬁahlothntcnnhomndnintonpnmmotarindmdbing the solution set of the swim. Thus if the 31m in oomlialitolmtI it will have inﬁnitelyr many solutions. EIEHDIﬁE SET 2.! 11. The given matrix corresponds to the system 31-53: +325=-2 £3 +ltrn= T n+5¢5= '3 where theequeﬂanmrreapmdingtntheeemmhnnbeenomltmd. Sohdngtheaeaquntinnn For the leading variables {\$1.23. and ml} in ﬁrmer-{the free variahlm {an and \$5.} reunite in xlzui+ﬁwg—3:rs, agar-£115 ends. =E—5m5. Thamigzﬁngarhitmyvelueetnmg and rhthe salutionset can be represented bythe parametric equations :1 =—2+ﬁe—-3t, mgr-e, zazT—4t, sures-ﬁt, 35:: where —ea<:e,t<:ee. Theoerreapnndingveetorﬁnrmin (11,1333, 54,175) = [—2. I], T, 3, El} + \$5,], 0,1}.0} +t{—3. 11-4. -5. I) an. The augmented matrix ofthe new is 3 2 —l —15 5 3 2 I} 3 1 3 11 ﬁ —4 ‘2 3D and theredueedmwechelenﬁnrmnfthiemnrrixie 1 D I] I} I] 1 I] {II I} U 1 I} i} I] I} 1 Thus the system is inconsistent. 45- The ahIla-neuter! mntrlx enha system n l 1 T —T 2 3 IT -16 1 3 a“ + 1 3a This redrawn to 1 l T —T «D I :t —2 n a {13—9 3n+9 Thelnntmwenrraepondntn£n2-9)z=3n+9. Ifn=m3thieheenmenﬂ=ﬂmnd theme-temwill have inﬁnitely many nolntiene. Ife-S. then the teat waterway-end! to {1:13; the system in inconsistent. Hnﬁiithenz=ﬁiﬂ=ﬁgmifmmhnekaublﬂtuﬂnmymdzareunlquely determinednnwellgthenyetemhaeexerthrnneeelnﬁnn. D8. [3} The. FerexnmpleLE ﬂmheredueedtneither Eilnr [111 (1". {b} Fahe.Theredueedrnwechelenﬁnmofnmatrixleuntqnn. {c} MnThenppmneenfnmwnfmmnmthnttheremenmeredundnncyinthe system.Bmthemnainingeqnnthmmnyheinennsietent,hnwmeﬂynnemlutm,whm inﬁnitelymnnarnelntiunn. ﬁlleftheeeere possible. . . nfthe d FﬂmﬁTheremnyheredtmdnneymtheaynbethrmpleJheeystemmnﬂstmg {} eqnnﬁomz+y-l,2e+£y=2,end3r+3y-3haeh1ﬂnitehrmnnyeulntlm. FL {I} [fag-Emthnnthamdmthnmbeamplhhedaaﬁﬂlwa: a b 1 5 1 g .5] 1 [I] [a d]"[c d]-'[n “592]" u 1 A u 1 Ifn=ﬂ.thenbg£ﬂmdcyéﬂ.mtheroducﬁun¢anhewﬁoduutasﬂnﬂows: u 1 c 1 5 1 g [I a] [c all—“[11 3*[0 1]”[0 11" u 1 {b} If [:3 1:11.11Innarad-11111:!£111[“1l ﬁlthﬁﬁthemjﬂSMQPBrﬂmaﬂthBW matrix 35’; wmmdmitmamatmonhemﬂ 5' fl mdﬁnmthisithﬂmthat themtamhastheuniquamluﬁonx=fﬂy=h CHAPTER 3 Matrices and Matrix Algebra EIEFICISE SET 3.1 4. {all masize2x4,ATl:-assimedx3. (b) E"12=211IIF21=‘I ('3) 5i: = 1 ifanﬂ on]? if {i1j1' {2:2} 01' {2:3} {'5} HUT} '[1 4 3] {a} 0293?}: meu 2 22. {a} uTv=-[3 ~—4 55[T]=ﬁ—23+U=J22 I] a a 21 u (11] mi": --1 [2 1- n]: —s —23 a 5 1n 3:. u {c} triuvTI=E-'EE+D=+2'2=11TV 3 {d} vTu=[2 1 n] --1 -ﬁ—13+ﬂ-—22=uTv 5 T {e} 111211115") ntrhru'r] =u-v =v'u-uT1r=1r 1.1:- —22 ee+ee eons 15+ 113 11 . . _. au+el some eu+25 e1 53 e5 34. {a} Theentneeet‘themetan+J- ”+9 “+12 45+“ 31 n, m repreeent the 15+e w+1u 35-1-11 23 to 4.1 total uniteeeldineeehetthe categories during the InmtheeﬂfhleyendJune. Fur example, thetetelnumberofmedimn reimeete enldweeMu-i-Jgg =4ﬂ+ It]: 51}. 15 2'1 er: :1 t' 15 3 53 2.1 T 311 M June eelee in each of the eetegeﬁee. Note that June sales were [use than May eelee in each cage; thrue these entries represent decreeem {h} The eutriee of the matrix M -J - repreeent the difference between May and l 45 W 180 1 -. 311 m Ill 11:!) {e} Let: [:J.‘I11enthemmponentenfﬂx- 13 65 t5 [1 = 122 representthetotelnum— 15 4D 35 I 90 her {ell elem} nfehirte,jeene.euite,endretneeeteeoldinMey. {d} Let3r=ll 1 1 1]. Thentheeempenenteel' {15 Ill] T5 3D 3|] 4|] 1? 65 45 15 4C} 35 1PM=[1 1 1 1] =[1ee 195 2115] mmlhetﬂtHleEIDfsmnﬂ.medium1mdlargeitemsmldlnMay. 4.551115 1 11111 _ 311110111 11111 {e} The 91:1thny [1 1 1 1] 12 554,5 Hzﬂ 1 1 1] 1:! =493 representethe 1511135 1 911 hotel mmher il'iterne {eﬂeleeeenrleetegorleeheld in May. 1:19.13] False.FnrezempleJt‘Aieitxﬁenchitx2,thenA.BendBAerehethdeﬁned. {1:} True. lfABendBAerehr-thdeﬁnedendﬁismxmtheanuetbenxm;thueABie mxmnndBAienxrt.If.inaddition,ﬂﬁ+3ﬂiedeﬁnedthenﬂ3mdﬁdmuetheme the same size. he. :11 == 11. [e] ‘I'rlne+ Ben:- the column rule1 1151:1113] = 1113(3). Thu! if B hes a when 01' meme. then .113 will have a column at" :eroe. (d1 Fame- Pm me. [3 1H? 3] - [i 11+ {13} nut. IfAienxn-t.thenATiemxn.AATienxn.endATAientxm. ThueATAend AAT hethereequare metrieee, endentrIIATJl} endtrHAT} ere hnthrleﬁned. [n Fl“.Humdvmlgnmmmﬁhenurvieenﬂxnmﬂm P21 Since Ax = m1e1{A}+ 3202b!) + - - . + :rnenLﬂ. the linear eyetem Ax - h is eqﬁvelent to nettle} + mental} + ~- +mnch} - b Thmtheeyetemdx=hieenneietentifendnnlyifthemterheanhe pressed ' mmhlnattenufthe column vectors of A. ex as e hneer ...
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