S.3 Math Matrices Caltec S.3

MATRICES Summary: 1.Amatrixisabracketwithnumbersinrowsandcolumns.Thusand arematrices. 2.Theorderofamatrixwithmrowsandncolumnsiswrittenasand iscalledanmatrix. 3.Thenumbersinamatrixarecalleditselementsorentries. 4.(i)Toaddandsubtractmatricesofthesameorder,addandsubtract correspondingelements (ii)Twomatricesareequaliftheircorrespondingelementsareequal (iii)Ascalarkmultipliedbyamatrix istreatedasfollows: (iv)Matrixmultiplicationistreatedasfollows: Matrixproduct . MatrixproductABcanbedoneifthenumberofcolumnsinAisequalto thenumberofrowinB. Ifa2´5matrixismultipliedbya5´3matrix,thentheresultingmatrixhas theouterdimensions(Thenewmatrixisoforder2´3) 5.AnidentitymatrixIisamatrixwithonesalongthemajordiagonaland zeroselsewhere.Thusa2´2identitymatrixisgivenby 6.Ifmatrix ,then: (i)DeterminantofA(DetA) (ii)AdjointofmatrixA= (iii)TheinverseofA, (iv)Theproduct (v)Amatrixmultipliedbyanidentitymatrixremainsunchanged 7.Asingularmatrixistheonewhosedeterminantiszeroandthushasno inverse. EXAMPLES: 1.Ifmatrix (a)StatetheorderofmatrixA (b)Determinethe: (i)determinantofA (ii)inverseofA 2.Giventhatmatrix and find: (i)P+Q(ii)QR(iii)3P2Q+R(iv)PQ(v)QP(vi)QRP (vii)(viii)(ix) (x)3P2IwhereIisa22identity matrix 3.Giventhatmatrix and finddet(AB) 4.Giventhatmatrix and find 5.Findtheorderoftheresultingmatrixwhena34matrixismultipliedby a 45matrix 6.Ifmatrix and (i)determinetheorderofmatrixAB (ii)findmatrixAB 7.Ifmatrix and (i)determinetheorderofmatrixR (ii)findmatrixR 8.GiventhematrixequationAY=B,usematrixinversionmethodtofind: (i)matrixY(ii)matrixA 9.Ifmatrix findmatrixBsuchthat 10.Ifmatrix findmatrixAsuchthat 11.Ifmatrix and determine: (i)theorderofmatrixR (ii)matrixR 12.Ifmatrix findmatrixAsuchthat whereIisa2 2identitymatrix. 13.Ifmatrix and findthevaluesxandy suchthat3A=B 14.Findthevaluesofaandbsuchthat 15.Findthevaluesofkandnsuchthat 16.Findthevaluesofxandysuchthat 17.Findthevaluesofxandysuchthat 18.Giventhatmatrix findthevalueofsuchthat whereIisa22identitymatrix 19.Giventhatmatrix and findthepossible valuesofxsuchthatAB=BA 20.Findthevaluesofxforwhichthematrix hasnoinverse 21.Findthevaluesofxforwhichthematrix issingular 22.Findthevaluesofxforwhichthematrix issingular 23.Giventhatmatrix findthevaluesofsuchthatthe matrix issingular,whereIisa22identitymatrix 4.Giventhat ,showthat 7.Giventhatm ,findand 1.If andisaidentitymatrix,provethat 2. 6.Giventhematrices and ,findmatrixMsuch that whereIisa22identitymatrix SOLUTIONTOSIMULTANEOUSEQUATIONSBYMATRIXMETHOD Summary: Thefollowingstepsapplyinsolvingsimultaneousequationusingmatrix method: (i)Writetheequationsinmatrixform (ii)Findtheinverseofthe2´2matrix (iii)Premultiplybothsidesofthematrixequationbytheinversematrix EXAMPLES: 1.Usematrixmethodtosolvethefollowingsimultaneousequations: 2.Findtheinverseofhencesolvethesimultaneousequations 3.Tombought3pensand2booksatShs4,800.Bobbought5pensand4 booksfromthesameshopatShs9,000. (i)Formtwoequationstorepresenttheaboveinformation (ii)Usematrixmethodtofindthecostofeachpenandthatofeachbook (iii)HowmuchwouldBenpayfor10pensand6books 4.Shs4000canbuy10bansand5cakesor4bansand10cakes. (i)Formtwoequationstorepresenttheaboveinformation (ii)Findbymatrixmethodthecostofeachbanandthatofeachcake. MATRIXWORDPROBLEMS 1.Tom,BobandBenwenttoasupermarketforshopping. Tombought3pensand5booksand4rulers Bobbought4pensand3booksand2rulers Benbought6pensand3rulers ThecostofapenisShs500,abookisShs800andarulerisShs1500. (a)Writedown: (i)a3´3matrixfortheitemsboughtbythethreeboys. (ii)a3´1costmatrixforeachitem (b)Usematrixmultiplicationtofindtheamountofmoneyspentbyeach boy 2.Inaswimmingcompetition,7pointswereawardedforeachfirst-place finish, 4pointsforsecondand2pointsforthird. Senioronehad4firstplacefinishes,7secondplacefinishesand3third placefinishes. Seniortwohad8firstplacefinishes,9secondplacefinishesand1third placefinish. Seniorthreehad10firstplacefinishes,5secondplacefinishesand3third placefinishes. Seniorfourhad3firstplacefinishes,3secondplacefinishesand6third placefinishes. (a)Writedown: (i)a4´3matrixforthenumberoffinisheseachclasshad. (ii)a3´1matrixforthepointsawardedforeachfinish (b)Usematrixmultiplicationtodeterminethewinnerofthecompetition 3.ShopsA,B,C,andDorderedforballs,batsandglovesasfollows: BallsBatsGlove s ShopA703050 ShopB602025 ShopC401510 ShopD504030 TheballscostShs5,000each,batsShs3,000eachandglovesShs2,000 each (a)Writedown: (i)a43matrixfortheitemsorderedbyeachshop. (ii)a31costmatrixforeachitem (b)Bymatrixmultiplication,findthetotalcostoftheitemsforeachshop (c)Ifthesupplierhadtopayataxof20%ofthecostoftheitemssold,find hisexpenditureontheorder. EER: 1.GiventhatIisanidentitymatrixoforder2×2andmatrix findmatrixB=A+2I 2.Findtheinverseofmatrix 3.Usematrixmethodtosolvethesimultaneousequations: 8.AhotelrentsdoubleroomsatShs40,000perdayandsingleroomsat Shs25,000perday.If14roomswererentedonedayforatotalofShs 470,000 (i)Formtwoequationstorepresenttheaboveinformation (ii)Findbymatrixmethodhowmanyroomsofeachkindwererented. 4.Inthemorning,5breadsand8cakeswerebought. Intheafternoon,7breadsand6cakeswerebought. ThecostofabreadisShs4000andacakeisShs1200 (a)Writedown: (i)a2×2matrixfortheboughtitems (ii)a2×1costmatrixforeachitem (b)Usematrixmultiplicationtofindtheexpenditureineachcase. 17.Giventhatmatrix findthevaluesofxandysuchthat whereIisa22identitymatrix 4.Giventhat and find: (i)theinverseofP. (ii)matrix 5.Findthevaluesofxforwhichthematrix hasnoinverse 19.Findthevaluesofxforwhichthematrix issingular 19.Findthevaluesofxforwhichthematrix issingular 19.Findthevaluesofxforwhichthematrix issingular 19.Findthevaluesofxforwhichthematrix issingular 21.Giventhatmatrix findthevaluesofksuchthatthe matrix issingular,whereIisa22identitymatrix 21.Giventhatmatrix findthevaluesofsuchthatthe matrix issingular,whereIisa22identitymatrix 16.Findthevaluesofxandysuchthat 2.Giventhatmatrix and findthe valueofxforwhichthedeterminantofRis2 21.Giventhatmatrix findthevaluesofxforwhichthe determinantofPis6 6.Shs244,000canbuy5bansand6cakes,whileShs356000canbuy7 bans and9cakes.Findbymatrixmethodthecostofeachbanandthatofa cake. 7.Findthevaluesofyforwhichthematrix issingular 8.Giventhatmatrix findthevalueofsuchthat whereIisa22identitymatrix 9.Giventhematrix ,findthevaluesofxandysuchthat whereIisa22identitymatrix. 9.Giventhematrix ,findthevaluesofxandysuchthat 10.BobandBenwenttoasupermarketforshopping. Bobbought2kgofsugar,4barsofsoap,5counterbooksandonebottle ofcookingoil. Benbought5kgofsugar,3barsofsoapandadozenofcounterbooks. ThecostofsugarperkgwasShs1,500,abarofsoapwasShs1,000,a counterbookwasShs3,000andabottleofcookingoilwasShs2,000. (a)Writedown: (i)a24matrixfortheitemsboughtbythetwopeople. (ii)a41costmatrixforeachitem (b)Calculatethe: (i)expenditureofeachpersonbymatrixmultiplication (ii)totalexpenditureofbothBobandBen (c)HowmuchdidBenspendthanBob 3.AcharityorganizationdonatedBallpens,exercisebooks,graphbooks andtablebookstoseniorfour,threeandtwoClassesofaschoolasbelow; Seniorfourstudentsgot2ballpens,12exercisebooks,3graphbooksand 1tablebookeach. Seniorthreestudentsgot2ballpens,8exercisebooks,1graphbooksand 1tablebookeach. Seniortwostudentsgot1ballpens,6exercisebooksand1tablebook each Thereare100studentsinseniorfour,120inseniorthreeand130students inseniortwo. Theorganizationboughttheitemsatthefollowingrates: BallpensatShs500each,ExercisebooksatShs1500each,graphbookat Shs2000eachandtablebooksatSh.6000each. (a)Writedown (i)1´3matrixforthenumberofstudents. (ii)3´4matrixfortheitems (iii)4´1costmatrix. (b)Bymatrixmultiplication,determinethe (i)numberofitemsofeachtypedistributed. (ii)totalamountspentbytheorganizationinacquiringtheitems. (c)Iftheorganizationhadtopay5%VATontheitemsbought,determine thetotalamountspent. 10.Ifmatrix findmatrixAsuchthat 9.Ifmatrix findmatrixAsuchthat 15.If ,findxandy 16.Giventhat ,findxandy. 1.Giventhat , and ,find; 2.i) 3.ii) Givethat , and ,findmatrixTsuch 4.If ; a)Determine;i) ii) b)identifymatrix 12.FourSecondaryschoolsfootballteamsofNtareH.S,LayibiCollege,MvaraS.S andKitendeS.Squalifiedforafootballtournament,whichwasplayedintwo roundswithotherteams. Firstround NtareH.Swonthreematches,drewoneandlostonematch. Layibicollegewontwomatches,drewoneandlosttwomatches …