MVR4 I)lsysquiz=6"%&falseGE\[l&%(quiz1_2G:F$6+%"iG%"jG%"kG%&LIMITG%(percentG%$ansG%% ans1G%%quesG%&qlistGF$F$C6-%%withG6#%'linalgG>8'"#5>%&scoreG""!>%%flagGF=-%&pri ntG6#%Z~~~~~~~Quiz~on~Consistent~and~Inconsistent~Linear~SystemsG-FA6#%jn~~~--- ---------------------------------------------------------G-%'printfG6$%UThis~qu iz~consists~of~%d~multiple~choice~questions.|+GF9-FH6#%fnYou~will~be~given~two~ chances~to~find~the~answer.~Type~the|+G-FH6#%jnanswer~followed~by~;~and~press~e nter.~You~may~quit~at~any~time|+G-FH6#%inby~typing:>exit~at~the~Maple~sign~but~ no~score~will~be~given.|+G-%&blankG6#"""-FH6#%hnType~;~and~Press~Enter~to~conti nue~or~exit;~to~quit~the~test|+G>%)responseG-%)readstatGF$@$0Ffn%%exitGC7>%$num GFW>%-numquestionsGFW>8,<">%&wlistG-%'vectorG6#"#:>8)61%"aG%"dGF^p%"cGF]pF_pF]p F_p%"bGF`pF_pF`pF_pF^pF`p>%&countGFW?(F$FWFWF$31FbpF9FjnC$>8+,&-%$modG6$-%%rand GF$FioFWFWFW@$4-%'memberG6$FhpFboC%-%,question1_2G6$Fhp&F[p6#Fhp>Fbo-%&unionG6$ Fbo<#Fhp>Fbp,&FbpFWFWFW>F?FWFT-FH6#%QType~";"~and~Press~Enter~to~get~the~final~ score|+GFgnFT@$/FfnF[oC$-FA6#%AWARNING~!!!~You~exit~too~quicklyGFT-FH6$%ATotal~ number~of~questions~=~~%d|+G,&F`oFW!""FW-FH6$%ANumber~of~CORRECT~answers~=~~%d| +GF<-FH6%%NThe~percentage~of~correct~answers~=~%.1f%c~|+~G,$*&F8(Ffs@+1"#!*F]t-FA6$%MOutstanding~Performance.~Percentage~score~isGF ]t1"#!)F]t-FA6$%FGood~performance.~Percentage~score~isGF]t1"#qF]t-FA6$%IAverage ~Performance.~Percentage~score~isGF]t1"#gF]t-FA6$%OBelow~average~Performance.~P ercentage~score~isGF]t-FA6$%LNot~a~good~performance.~Percentage~score~isGF]tFT@ $0,&FF[pFgn@%5/F[p%$yesG/F[p%$YESGC&-FA6#%?INCORRECTLY~ANSWERED~QUE STIONSG-FA6#%?******************************GFT?(8$FWFW,&F^oFWF_sFW%%trueGC&-FH 6$%6Question~Number~%d~:~G&&Feo6#F\w6#""$-%/questionset1_2G6#&FdwFV>8*Fgn@%/F]x &Fdw6#""#-FA6#%AGOOD.~This~is~the~correct~answerGC$-FA6#%FIncorrect~answer.~The ~correct~ans~is:G-FA6#F`x-FA6#%$BYEGFT-FH6#%gn~~Thank~you~for~using~the~automat ed~testing~system~of~ILAT.|+GFT-FH6#%F~~To~Return~to~ILAT~menu~type~quit;~|+G-F H6#%I~~To~take~another~quiz,~type~quiz1_2();|+G-FH6#%=~~To~exit~press~;~and~ent er|+G>F[pFgnF$6*F`oFF9F:>FF ?F=-FA6#%gn~~~~~~~~~~~~Quiz~on~Gauss,~Gauss-Jordan~and~BacksubstitutionG-FA6#%j n~~~~~~~~~~~----------------------------------------------------GFGFKFNFQFTFX>F fnFgn@$FjnC7>F^oFW>F`oFW>FboFco>Feo-Fgo6#"#?>F[p66F]pF]pF^pF`pF^pF`pF^pF]pF_pF_ pF_pF]pF_pF`pF^pF]pF`pF_pF`pF_p>FbpFW?(F$FWFWF$FdpC$>Fhp,&-F[q6$F]qFc[lFWFWFW@$ F`qC%-%,question1_5GFgq>FboF[r>FbpF`r>F?FWFT-FH6#%EPress~;Enter~to~get~the~fina l~score|+GFgnFT@$FfrC$FhrFT-FH6$%D~~~Total~number~of~questions~=~~%d|+GF^s-FH6$ %D~~~Number~of~CORRECT~answers~=~~%d|+GF<-FH6%%Q~~~The~percentage~of~correct~an swers~=~%.1f%c~|+~GFfsFisFT@$FjnC$>F]tFfs@+F_tFatFdtFftFitF[uF^uF`uFcuFT@$FguC% -FH6#%`oWould~you~like~to~try~incorrect~questions~again~?~answer~yes;~or~no;|+G >F[pFgn@%F_vC&FevFhvFT?(F\wFWFWF]wF^wC&F`w-%/questionset1_5GFjw>F]xFgn@%F_xFcxC $FgxFjxF\yFTF_yFT-FH6#%DTo~Return~to~ILAT~menu~type~quit;~|+G-FH6#%GTo~take~ano ther~quiz,~type~quiz1_5();|+G-FH6#%;To~exit~press~;~and~enter|+G>F[pFgnF$F\z%(q uiz1_1G:F$F)F$F$C6F4>F9F:>FF?F=-FA6#%T~~~~~~~~~~~~~~~~~Quiz~on~Examples~of~ Linear~SystemsG-FA6#%X~~~~~~~~~~~~~~~----------------------------------------GF GFKFNFQFT-FH6#%jn<~Type~;~and~press~Enter~to~continue~or~exit;~to~quit~the~test |+G>FfnFgn@$FjnC7>F^oFW>F`oFW>FboFco>FeoFfo>F[p61F`pF]pF^pF`pF_pF_pF^pF_pF`pF_p F`pF`pF^pF_pF`p>FbpFW?(F$FWFWF$FdpC$>FhpFip@$F`qC%-%,question1_1GFgq>FboF[r>Fbp F`r>F?FWFTFbrFgnFT@$FfrC$FhrFTF[sF`sFcsFT@$FjnC$>F]tFfs@+F_tFatFdtFftFitF[uF^uF `uFcuFT@$FguC%Fh]l>F[pFgn@%F_vC&FevFhvFT?(F\wFWFWF]wF^wC&F`w-%/questionset1_1GF jw>F]xFgn@%F_xFcxC$FgxFjxF\yFT-FH6#%enThank~you~for~using~the~automated~testing ~system~of~ILAT.|+GFTFby-FH6#%I~~To~take~another~quiz,~type~quiz1_1();|+GFhy>F[ pFgnF$F\z%(quiz1_3G:F$F)F$F$C6F4>F9F:>FF?F=-FA6#%S~~~~~~~~~~~~~~~~~Quiz~on~ Elementary~Row~OperationsG-FA6#%V~~~~~~~~~~~~~~~------------------------------- -------GFGFKFNFQFTFX>FfnFgn@$FjnC7>F^oFW>F`oFW>FboFco>FeoFfo>F[p61F_pF`pF]pF^pF ^pF^pF`pF_pF^pF`pF`pF_pF^pF]pF`p>FbpFW?(F$FWFWF$FdpC$>FhpFip@$F`qC%-%,question1 _3GFgq>FboF[r>FbpF`r>F?FWFTFbrFgnFT@$FfrC$FhrFTFi\lF\]lF_]lFT@$FjnC$>F]tFfs@+F_ tFatFdtFftFitF[uF^uF`uFcuFT@$FguC%Fh]l>F[pFgn@%F_vC&FevFhvFT?(F\wFWFWF]wF^wC&F` w-%/questionset1_3GFjw>F]xFgn@%F_xFcxC$FgxFjxF\yFTF_yFTFe^l-FH6#%GTo~take~anoth er~quiz,~type~quiz1_3();|+GF[_l>F[pFgnF$F\z%(quiz1_4G:F$F)F$F$C6F4>F9F:>FF? F=-FA6#%hn~~~~~~~~~~~~~~Quiz~on~Matrix~Representation~of~Linear~SystemsG-FA6#%j n~~~~~~~~~~~~---------------------------------------------------GFGFKFNFQFT-FH6 #%inType~;~and~Press~Enter~to~continue;~or~exit;~to~quit~the~test|+G>FfnFgn@$Fj nC7>F^oFW>F`oFW>FboFco>FeoFfo>F[p61F]pF`pF_pF^pF`pF^pF`pF^pF`pF^pF]pF_pF_pF_pF^ p>FbpFW?(F$FWFWF$FdpC$>FhpFip@$F`qC%-%,question1_4GFgq>FboF[r>FbpF`r>F?FWFTFd\l FgnFT@$FfrC$FhrFTF[sF`sFcsFT@$FjnC$>F]tFfs@+F_tFatFdtFftFitF[uF^uF`uFcuFT@$FguC %Fh]l>F[pFgn@%F_vC&FevFhvFT?(F\wFWFWF]wF^wC&F`w-%/questionset1_4GFjw>F]xFgn@%F_ xFcxC$FgxFjxF\yFTF_yFTFby-FH6#%I~~To~take~another~quiz,~type~quiz1_4();|+GFhy>F [pFgnF$F\zF$, I/questionset2_1:6#%"nG6,%"AG%#A1G%#A2G%#A3G%#A4G%#B1G%#B2G%#B3G%#B4G%"BGF$F$C& @$/F?F=C$-FH6$%4QUESTION~NUMBER~%d|+GFbpFT-FH6#%Y|+**************************** **************************|+G@A/9$FWC+-FH6#%.The~matrices|+G>F\w-%'matrixG6#7#7 %FW,&%"xGFW%"yGFW%"zG>8--F\\m6#7#7%Fa\mFc\m""'-FA6$/F\w-%&evalmGFew/Fe\m-F_]m6# Fe\m-FH6#%.are~equal~if|+G-FH6#%/(a)~x=1,y=z=6|+G-FH6#%1(b)~x=1,y=5,z=6|+G-FH6# %2(c)~x=1,y=-5,z=6|+G-FH6#%1(d)~x=1,y=z,z=5|+G/Fe[mFbxC'-FH6#%QThe~product~of~t wo~upper~triangular~matrices~is|+G-FH6#%@(a)~an~upper~triangular~matrix|+G-FH6# %?(b)~a~lower~triangular~matrix|+G-FH6#%9(c)~a~symmetric~matrix~|+G-FH6#%8(d)~a ~diagonal~matrix~|+G/Fe[mFgwC+-FH6#%XThe~echelon~form~of~an~augmented~matrix~of ~a~system~of|+G-FH6#%>linear~equations~is~given~by|+G>F\w-F\\m6#7%7&FWF=FW%#a1G 7&F=FWFbx,&%#a2GFWF``m!"$7&F=F=F=,(%#a3GFWFc`m!"#F``mFW-FA6#F]]m-FH6#%TThen~the ~system~has~infinitely~many~solutions~if~~|+G-FH6#%8(a)~a1,a2,a3~arbitrary|+G-F H6#%0(b)~a1-a2-a3=0|+G-FH6#%0(c)~a3=2*a2+a1|+G-FH6#%0(d)~a3=2*a2-a1|+G/Fe[m""%C *-FH6#%VA~is~a~matrix~such~that~its~cube~is~equal~to~itself.|+G>F\w.F\w-FA6$%(T hat~isG/*$F\wFgwF\w-FH6#%&Then|+G-FH6#%C(a)~A~can~only~be~the~zero~matrix|+G-FH 6#%G(b)~A~can~only~be~the~identity~matrix|+G-FH6#%?(c)~A~can~only~be~invertible |+~G-FH6#%<(d)~A~to~the~power~15~is~A|+G/Fe[m""&C*Fg[m>F\w-F\\m6#7$F_\m7%Fbx,&F a\mFWFb\mF_sF[bm>Fe\m-F\\m6#7$Fi\m7%,&Fc\mFW!"%FWFedmF[bmF[]m-FH6#%<(a)~are~equ al~if~x=1,y=z=6|+G-FH6#%>(b)~are~equal~if~x=1,y=5,z=6|+G-FH6#%?(c)~are~equal~if ~x=1,y=-5,z=6|+G-FH6#%P(d)~are~not~equal~for~any~values~of~x,y,~and~z|+G/Fe[mFj \mC'-FH6#%TThe~product~of~an~nxr~matrix~A~and~rxm~matrix~B~is|+G-FH6#%3(a)~an~m xr~matrix|+G-FH6#%3(b)~an~nxm~matrix|+G-FH6#%1(c)~not~defined|+G-FH6#%0(d)~rxn~ matrix|+G/Fe[m""(C)-FH6#%TThe~(3,2)~entry~in~the~product~of~the~two~matrices|+G -FA6#/F\w-F_]m6#-F\\m6#7%7%FWFbxF=7%Fd`mFbxFW7%FgwFWFgw-FA6#/Fe\m-F_]m6#-F\\m6# 7%7$FWFbx7$Fd`mFbx7$FWFgcm-FH6#%,(a)~is~-23|+G-FH6#%+(b)~is~23|+G-FH6#%*(c)~is~ 6|+G-FH6#%+(d)~is~15|+G/Fe[m"")C'-FH6#%SWhich~one~of~the~following~statements~i s~correct?|+G-FH6#%R(a)~The~sum~of~two~nxm~matrices~is~an~nxm~matrix|+G-FH6#%U( b)~The~product~of~any~two~matrices~is~well~defined|+G-FH6#%[o(c)~Multiplication ~of~matrices,~whenever~defined,is~commutative|+G-FH6#%hn(d)~Addition~of~matrice s~whenever~defined~is~not~commutative|+G/Fe[m""*C'-FH6#%WWhich~one~of~the~follo wing~statements~is~not~correct?|+GFaim-FH6#%^o(b)~The~product~of~an~nxm~matrix~ and~an~mxn~matrix~is~well~defined|+G-FH6#%\o(c)~Multiplication~of~matrices,~whe never~defined,~is~commutative|+G-FH6#%Z(d)~Addition~of~matrices~whenever~define d~is~commutative|+G/Fe[mF:C'F^im-FH6#%jn(a)~Addition~of~matrices,~whenever~defi ned,~is~not~associative|+G-FH6#%T(b)~The~additive~inverse~of~a~matrix~is~not~un ique|+G-FH6#%N(c)~The~zero~matrix~is~the~additive~identity|+GFjim/Fe[m"#6C'F^im -FH6#%in(a)~The~product~of~a~matrix~with~a~scalar~is~always~a~scalar~|+G-FH6#%^ o(b)~The~product~of~a~matrix~with~a~scalar~is~a~matrix~of~same~size|+G-FH6#%[o( c)~The~product~of~a~matrix~with~a~scalar~is~not~always~defined|+G-FH6#%\o(d)~Th e~product~of~a~matrix~A~with~a~scalar~is~always~equal~to~A|+G/Fe[m"#7C*-FH6#%,T he~matrix|+G-FA6#/F\w-F_]m6#-F\\m6#7$7$FWFW7$F_sFW-FH6#%:commutes~with~the~matr ix|+G-FA6#/Fe\m-F_]m6#-F\\m6#7$7$F]pFW7$F_sF`p-FH6#%G(a)~if,~for~example,~a~=~2 ~~and~b~=~3|+G-FH6#%H(b)~if,~for~example,~a~=~3~~and~b~=~3~|+G-FH6#%G(c)~if,~fo r~example,~a~=~3~~and~b~=~2|+G-FH6#%G(d)~if,~for~example,~a~=~4~and~~b~=~5|+G/F e[m"#8C*-FH6#%EThe~value~of~a~for~which~the~matrix|+G-FA6#/F\w-F_]m6#-F\\m6#7$F a^nFd]n-FH6#%?is~a~solution~to~the~equation|+G-FA6#/,&*$Fa\mFbxFWFa\mFh`mF=-FH6 #%+(a)~is~-1|+G-FH6#%*(b)~is~1|+G-FH6#%*(c)~is~2|+G-FH6#%+(d)~is~-2|+G/Fe[m"#9C )-FH6#%2Given~the~matrix|+G-FA6#/F\w-F_]m6#-F\\m6#7$7$FWF_sFe]n-FH6#%8The~5-th~ power~of~A~is|+G-FH6#%;(a)~2~times~the~matrix~A~|+G-FH6#%;(b)~16~times~the~matr ix~A|+G-FH6#%;(c)~32~times~the~matrix~A|+G-FH6#%;(d)~10~times~the~matrix~A|+G/F e[mFioC,FdanFgan-FH6#%P(a)~The~matrix~A~is~a~solution~to~the~equation|+G-FA6#/, &*$Fa\mFgwFWFa\m!#;F=-FH6#%P(b)~The~matrix~A~is~a~solution~to~the~equation|+G-F A6#/,&FhcnFWFa\mFh`mF=-FH6#%P(c)~The~matrix~A~is~a~solution~to~the~equation|+G- FA6#/,&FhcnFWFa\mFedmF=-FH6#%P(d)~The~matrix~A~is~a~solution~to~the~equation|+G -FA6#/,&FhcnFWFa\m!")F=-FA6#%AAdd~more~questions~to~this~test!G-FH6#%G|+******* *****************************|+GF$F$F$, I/questionset2_2:F\jlF^jlF$F$C&@$F[[mC$F][mFTF`[m@AFd[mC'Fd^mFg^m-FH6#%:(b)~an~ invertible~matrix|+GF]_mF`_mFb^mC+Fe_mFh_m>F\wF\`mFi`mF[amF^amFaamFdamFgamFc_mC *Fe_mFh_m>F\w-F\\m6#7%F_`mFa`m7&F=F=FWFf`mFi`m-FH6#%[o(a)~The~system~has~infini tely~many~solutions~for~any~a1,a2,a3~~|+G-FH6#%Y(b)~The~system~has~a~unique~sol ution~for~any~a1,a2,~a3|+~G-FH6#%V(c)~The~system~has~a~unique~solution~if~a3-2* a2+a1=1|+G-FH6#%jn(d)~The~system~has~a~infinitely~many~solutions~if~a3-2*a2+a1= 1|+GFjamC)-FH6#%7The~matrix~A~given~by|+G>F\w-F\\m6#7%7%FWF=FW7%F=FWFbx7%F=F=FW Fi`m-FH6#%K(a)~is~an~example~of~an~elementary~matrix|+G-FH6#%I(b)~is~an~example ~of~a~symmetric~matrix|+G-FH6#%Q(c)~is~an~example~of~an~upper~triangular~matrix |+G-FH6#%P(d)~is~an~example~of~a~lower~triangular~matrix|+GFfcmC)Fegn>F\w-F\\m6 #7%7%FWF=Fgw7%F=FWF=F^hnFi`mF_hnFbhn-FH6#%U(c)~is~not~an~example~of~an~upper~tr iangular~matrix|+GFhhnFbemC*Fegn>F\w-F\\m6#7%7%FWFWF=FainF^hnFi`m-FH6#%2is~an~e xample~of|+G-FH6#%:(a)~an~elementary~matrix|+G-FH6#%7(b)~a~diagonal~matrix|+G-F H6#%J(c)~a~matrix~in~reduced~row~echelon~form|+G-FH6#%6(d)~symmetrix~matrix|+GF cfmC*Fegn>F\w-F\\m6#7&7&FWFWF=F=7&F=FWF=F=7&F=F=FWFW7&F=F=F=FWFi`mF[jnF^jnFajn- FH6#%A(c)~a~matrix~in~an~echelon~form|+GFgjnF[imC*Fg[m>F\wFjcm>Fe\mF`dmF[]mFfdm FidmF\emF_emF]jmC)Fegn>F\w-F\\m6#7%7%FWF=F=7%F=FWFedm7%F=FedmFWFi`mF_hnFbhn-FH6 #%H(c)~is~an~example~of~a~diagonal~matrix|+GFhhnF\[nC'F^im-FH6#%^o(a)~The~produ ct~of~two~elementary~matrices~is~an~elementary~matrix|+G-FH6#%_o(b)~The~transpo se~of~an~upper~triangular~matrix~is~lower~triangular|+G-FH6#%[o(c)~The~product~ of~two~symmetric~matrices~is~a~symmetric~matrix|+G-FH6#%[o(d)~The~sum~of~two~sy mmetric~matrices~is~not~a~symmetric~matrix|+GFg[nC.Fi\n>F\w-F\\m6#7%F\hnFainF^h nFi`m-FH6#%;then~the~matrix~A^(10)~is|+G>8%Fc]o>8&-F\\m6#7%7%FWF=F_sFainF^hn>F9 -F\\m6#7%7%FWF=F:FainF^hn>F]t-F\\m6#7%F^\oFainF^hn-FA6$%$(a)GF\^o-FA6$%$(b)GFj] o-FA6$%$(c)GF9-FA6$%$(d)GF]tFf\nC'-FH6#%PWhich~one~of~the~following~statements~ is~true?|+G-FH6#%T(a)~Any~two~matrices~always~commute.~That~is~AB=BA|+G-FH6#%`o (b)~The~product~of~two~upper~triangular~matrices~is~upper~triangular|+G-FH6#%\o (c)~Every~symmetric~matrix~is~equivalent~to~an~elementary~matrix|+G-FH6#%gn(d)~ The~transpose~of~a~symmetric~matrix~A~is~not~equal~to~A|+GF__nC)-FH6#%^oA~is~an ~nxn~matrix~which~is~not~row~equivalent~to~the~identity.~If|+G-FH6#%]oA~is~the~ coefficient~matrix~of~a~homogeneous~system~of~equations,|+G-FH6#%5then~the~syst em~has|+G-FH6#%1(a)~no~solution|+G-FH6#%7(b)~a~unique~solution|+G-FH6#%?(c)~inf initely~many~solutions|+G-FH6#%7(d)~none~of~the~above|+GFaanC)-FH6#%\oA~is~an~n xn~matrix~which~is~row~equivalent~to~the~identity.~If~A|+G-FH6#%[ois~the~coeffi cient~matrix~of~a~homogeneous~system~of~equations,|+GF]aoF`aoFcaoFfaoFiaoF_cnC) -FH6#%YPre-multiplying~a~2x2~matrix~A~by~the~elementary~matrix|+G-FA6#/%"EG-F_] m6#-F\\m6#7$7$FWF=7$FgwFW-FH6#%2is~equivalent~to|+G-FH6#%A(a)~multiplying~row~2 ~of~A~by~3|+G-FH6#%U(b)~multiplying~row~1~of~A~by~3~and~adding~to~row~2|+G-FH6# %A(c)~multiplying~row~1~of~A~by~3|+G-FH6#%E(d)~adding~row~1~of~A~to~row~2~of~A| +GF`enFcenF$F$F$, I/questionset2_3:F\jlF^jlF$F$C&@$F[[mC$F][mFTF`[m@KFd[mC'-FH6#%RThe~product~of~ two~invertible~matrices~is~always|+G-FH6#%4(a)~not~invertible|+G-FH6#%0(b)~inve rtible|+G-FH6#%7(c)~a~diagonal~matrix|+G-FH6#%9(d)~a~triangular~matrix|+GFb^mC' -FH6#%boIf~the~square~of~a~non~identity~nxn~matrix~A~is~itself,~then~A~must~be| +G-FH6#%0(a)~invertible|+G-FH6#%.(b)~identity|+G-FH6#%.(c)~singular|+G-FH6#%-(d )~a~or~c~|+GFc_mC(-FH6#%jnIf~the~coefficient~matrix~of~a~nonhomogeneous~system~ of~linear|+G-FH6#%Pequations~is~invertible,~then~the~solution~set|+G-FH6#%>(a)~ consists~of~two~elements|+G-FH6#%1(b)~is~infinite|+G-FH6#%<(c)~consist~of~one~e lement|+G-FH6#%/(d)~is~empty~|+GFjamC(Fhfo-FH6#%Nequations~is~singular,~then~th e~solution~set|+G-FH6#%/(a)~is~empty~|+GFagoFdgo-FH6#%8(d)~none~of~the~above~|+ GFfcmC'-FH6#%O~The~sum~of~two~invertible~matrices~is~always|+GF[fo-FH6#%.(b)~si ngular|+G-FH6#%0(c)~triangular|+GFiaoFbemC)Fegn>F\wFignFi`mF_hnFbhn-FH6#%H(c)~i s~an~example~of~a~singular~matrix|+G-FH6#%K(d)~is~an~example~of~an~invertible~m atrix|+GFcfmC'-FH6#%YIf~a~matrix~A~is~such~that~A~is~equal~to~its~cube,~then|+G -FH6#%?(a)~A~must~be~the~zero~matrix|+G-FH6#%B(b)~A~must~be~the~identity~matrix G-FH6#%;(c)~A~must~be~invertible|+~GFccmF[imC'-FH6#%FIf~a~matrix~A~has~an~inver se~B,~then|+G-FH6#%W(a)~B~is~not~unique~since~A~has~more~than~one~inverse|+G-FH 6#%T(b)~B~is~not~row~equivalent~to~the~identity~matrix|+G-FH6#%P(c)~B~is~row~eq uivalent~to~the~identity~matrix|+G-FH6#%B(d)~(a)~and~(b)~are~both~correct|+GF]j mC'-FH6#%VIf~a~matrix~A~is~row~equivalent~to~a~matrix~B,~then~|+G-FH6#%E(a)~B~i s~invertible~while~A~is~not~|+G-FH6#%I(b)~A~and~B~donot~share~same~properties|+ G-FH6#%G(c)~A~and~B~do~share~same~properties~|+G-FH6#%O(d)~A~is~invertible~whil e~B~is~not~invertible|+GF\[nC*Fegn>F\wFginFi`mF[jnF^jnFajnFdjnFgjnFg[nC*Fegn>F\ wF\[oFi`mF[jnF^jnFajnFc[oFgjnFf\nC'-FH6#%HThe~inverse~of~an~elementary~matrix~i s|+G-FH6#%A(a)~is~not~an~elementary~matrix|+G-FH6#%=(b)~is~an~elementary~matrix |+G-FH6#%<(c)~is~a~triangular~matrix|+G-FH6#%;(d)~is~a~symmetrix~matrix|+GF__nC 1-FH6#%EThe~inverse~of~the~diagonal~matrix:|+G>F\w-%%diagG6'FWFbxFd`mF[bm!"&Fi` m-FH6#%<(a)~is~the~diagonal~matrix|+G>F[p-F]^p6'F_sFh`mFgwFedmFgcm-FA6#F[p-FH6# %<(b)~is~the~diagonal~matrix|+G>F]x-F]^p6'F_s#F_sFbx#FWFgw#F_sF[bm#FWFgcm-FA6#F ]x-FH6#%<(c)~is~the~diagonal~matrix|+G>Fhp-F]^p6'FW#FWFbx#F_sFgw#FWF[bm#F_sFgcm -FAFiq-FH6#%<(d)~is~the~diagonal~matrix|+G>Fbo-F]^p6'FWFj_pF__pF\`pFa_p-FA6#Fbo FaanC'-FH6#%QWhich~one~of~the~following~statements~is~false?|+G-FH6#%Z(a)~The~p roduct~of~two~invertible~matrices~is~invertible|+G-FH6#%en(b)~The~inverse~of~a~ diagonal~matrix~is~a~diagonal~matrix|+G-FH6#%N(c)~Every~invertible~matrix~has~t wo~inverses|+G-FH6#%T(d)~The~inverse~of~a~symmetric~matrix~is~symmetric|+GF_cnC 'Fg_o-FH6#%P(a)~The~product~of~two~matrices~is~commutative|+G-FH6#%L(b)~The~inv erse~of~a~matrix~is~~not~unique|+G-FH6#%S(c)~The~transpose~of~a~symmetric~matri x~is~itself|+G-FH6#%X(d)~The~inverse~of~a~symmetric~matrix~is~not~symmetric|+G/ Fe[m"#;C'-FH6#%RWhich~one~of~the~following~statements~is~correct|+G-FH6#%^o(a)~ The~product~of~a~singular~and~a~nonsingular~matrix~is~singular|+G-FH6#%ao(b)~Th e~product~of~a~singular~and~a~nonsingular~matrix~is~nonsingular|+G-FH6#%Y(c)~Th e~product~of~two~singular~matrices~is~nonsingular|+G-FH6#%Y(d)~The~product~of~t wo~nonsingular~matrices~is~singular|+G/Fe[m"#C)-FH6#%jnGiven~an~nxn~matrix~A~which~is~row~equivalent~to~the~identi ty.|+GF[epF]aoF`aoFcaoFfaoFiao/Fe[mFc[lC)Faep-FH6#%hnIf~A~is~the~coefficient~ma trix~of~a~nonhomogeneous~system~of|+G-FH6#%@equations,~then~the~system~has|+GF` aoFcaoFfaoFiaoF`enFcenF$F$F$, I/questionset2_4:F\jlF^jlF$F$C&@$F[[mC$F][mFT-FH6#%F|+************************* **********|+G@AFd[mC)-FH6#%]oThe~coefficient~matrix~of~a~nonhomogeneous~system~ of~equations~is|+G-FA6#/F\w-F_]m6#-F\\m6#7%FjinF]hnF\hn-FH6#%Ethen~the~solution ~set~of~the~system|+GF^goFagoFdgoFggoFb^mC)Fffp-FA6#/F\w-F_]m6#-F\\m6#7%FjinF]h n7%F_sF=FbxFagpF^goFagoFdgo-FH6#%9(d)~canot~be~determined|+GFc_mC)-FH6#%jnThe~c oefficient~matrix~of~a~homogeneous~system~of~equations~is|+GFegpFagpF^goFagoFdg oFggoFjamC)Fbhp-FA6#/F\w-F_]m6#-F\\m6#7%FjinF]hn7%FWF=FbxFagpF^goFagoFdgoFggoFf cmC(Fi\nFegp-FH6#%hn(a)~is~invertible~since~it~is~row~equivalent~to~the~identit y|+G-FH6#%`o(b)~is~not~invertible~since~it~is~not~row~equivalent~to~the~identit y|+G-FH6#%K(c)~is~an~example~of~an~elementary~matrix|+G-FH6#%Q(d)~is~an~example ~of~an~upper~triangular~matrix|+GFbemC(-FH6#%[oIf~the~coefficient~matrix~of~a~n on-homogeneous~system~of~linear|+GF[hoF^hoFagoFdgoFahoFcfmC'FehoF[foFhhoF[ioFia oF[imC'Fh\pF[]pF^]pFa]pFd]pF]jmC1Fh]p>F\wF\^pFi`mF`^p>F[pFd^pFf^pFh^p>F]xF\_pFb _pFd_p>FhpFh_pF^`pF_`p>FboFc`pFe`pF\[nC'Fh`pF[apF^apFaapFdapFg[nC)FhdpF[epF]aoF `aoFcaoFfaoFiaoFf\nC)FaepF[epF]aoF`aoFcaoFfaoFiaoF__nC'-FH6#%WThe~inverse(AB)~o f~two~invertible~matrices~A~and~B~is|+G-FH6#%>(a)~is~inverse(A)*inverse(B)|+G-F H6#%>(b)~is~inverse(B)*inverse(A)|+G-FH6#%4(c)~is~inverse(BA)|+G-FH6#%4(d)~does ~not~exist|+GFaanC(Fi\n-FA6#/F\w-F_]m6#-F\\m6#7%7%FWF_sFW7%FbxFWFgw7%F[bmF_sFgc m-FH6#%H(a)~is~an~example~of~a~singular~matrix|+G-FH6#%J(b)~is~an~example~of~a~ elementary~matrix|+G-FH6#%K(c)~is~an~example~of~a~nonsingular~matrix|+G-FH6#%I( d)~is~an~example~of~a~symmetric~matrix|+GF_cnC.-FH6#%4Given~the~matrices|+G-FA6 #/F\w-F_]m6#-F\\m6#7$Fd]n7$F=FW-FA6#/Fe\m-F_]m6#-F\\m6#7$F`coFd]n-FH6#%BThe~inv erse~of~the~product~AB~is|+G-FH6#%0(a)~the~matrix|+G-FA6#/%"CG-F_]m6#-F\\m6#7$F _bn7$F_sFbx-FH6#%0(b)~the~matrix|+G-FA6#/Fa_q-F_]m6#-F\\m6#7$Fd]nF\hm-FH6#%0(c) ~the~matrix|+G-FA6#/Fa_q-F_]m6#-F\\m6#7$F_bnF\hm-FH6#%0(d)~the~matrix|+G-FA6#/F a_q-F_]m6#-F\\m6#7$Fd]nFg_qF`enFcenF$F$F$, I/questionset2_5:F\jlF^jlF$F$C&@$F[[mC$F][mFTF`[m@KFd[mC(-FH6#%^oThe~determinan t~of~a~4x4~triangular~matrix~whose~diagonal~elements|+G-FH6#%6are~-2,~3,1~and~2 ~is|+G-FH6#%'(a)~4|+G-FH6#%((b)~-6|+G-FH6#%((c)~12|+G-FH6#%*(d)~-12~|+GFb^mC)-F H6#%KThe~determinant~of~the~matrix~A~given~by:|+G>F\w-F\\m6#7&7&FbxFWFWFgcm7&F= FWFbxFgw7&F=FbxFgwF\im7&F=FbxF[bmFj\mFi`m-FH6#%((a)~36|+G-FH6#%((b)~0~|+GF\cq-F H6#%)(d)~18~|+GFc_mC)-FH6#%VThe~cofactor~of~the~entry~5~in~the~matrix~A~given~b y|+G>F\w-F\\m6#7%F\hnF]hn7%F=FgcmFWFi`m-FH6#%)(a)~~2~|+G-FH6#%)(b)~-2~|+G-FH6#% ((c)~~1|+G-FH6#%((d)~-1|+GFjamC(-FH6#%\oThe~determinant~of~a~5x5~triangular~mat rix~is~4~and~its~diagonal|+G-FH6#%Melements~are~-2,2,1,-1,x.~The~value~of~x~is| +GFfbq-FH6#%((b)~-1|+G-FH6#%'(c)~1|+G-FH6#%)(d)~-4~|+GFfcmC2-FH6#%7If~for~a~2x2 ~matrix~A|+G>F\w-F\\m6#7$7$F]pF`p7$F_pF^pFi`m-FH6#%^othe~cofactor~of~each~entry ~is~equal~to~the~entry~itself,~then~A~is|+G>Fj]o-F\\m6#7$Fdgq7$,$F`pF_sF]p>F\^o -F\\m6#7$7$F]p,$F_pF_s7$F^hqF^p>F9Fagq>F]t-F\\m6#7$7$F^pFdhqF]hq-FH6#%.(a)~give n~by|+G-FA6#Fj]o-FH6#%.(b)~given~by|+G-FA6#F\^o-FH6#%.(c)~given~by|+G-FA6#F9-FH 6#%.(d)~given~by|+G-FA6#F]tFbemC)-FH6#%jnThe~values~of~x~for~which~the~followin g~matrix~is~singular~are|+G>F\w-F\\m6#7%7%Fa\mFWF=7%F=Fa\mFW7%F_sFWFbxFi`m-FH6# %>(a)~is~singular~are~|fr1,-1/2|hr|+G-FH6#%?(b)~is~singular~are~|fr-1,-1/2|hr|+ G-FH6#%>(c)~is~singular~are~|fr-1,1/2|hr|+G-FH6#%=(d)~is~singular~are~|fr1,1/2| hr|+GFcfmC'-FH6#%aoIf~A~is~2x2~invertible~matrix,~then~the~determinant~of~the~a djoint(A)|+G-FH6#%:(a)~is~equal~to~1/det(A)|+G-FH6#%9(b)~is~equal~to~-det(A)|+G -FH6#%8(c)~is~equal~to~det(A)|+G-FH6#%4(d)~is~equal~to~-1|+GF[imC(-FH6#%_oIf~th e~determinant~of~a~4x4~matrix~A~is~-3,~then~the~determinant~of|+G-FH6#%+of~2*A~ is|+G-FH6#%((a)~-6|+G-FH6#%'(b)~6|+G-FH6#%((c)~48|+G-FH6#%)(d)~-48|+GF]jmC(-FH6 #%_oIf~a~6x6~matrix~A~is~such~that~one~row~is~a~multiple~of~another~row|+G-FH6# %Fthen~the~determinant~of~the~matrix~A|+G-FH6#%'(a)~6|+G-FH6#%'(b)~1|+G-FH6#%'( c)~0|+G-FH6#%((d)~-6|+GF\[nC(-FH6#%^oIf~the~determinant~of~a~matrix~A~is~0,~the n~the~homogeneous~system|+G-FH6#%Sof~linear~equations~whose~coefficient~matrix~ is~A|+G-FH6#%B(a)~has~infinitely~many~solution|+G-FH6#%F(b)~has~a~unique~nontri vial~solution|+G-FH6#%C(c)~has~a~unique~trivial~solution|+G-FH6#%5(d)~is~incons istent|+GFg[nC(-FH6#%jnIf~the~determinant~of~a~matrix~A~is~0,~then~the~nonhomog eneous|+G-FH6#%insystem~of~linear~equations~Ax=b~whose~coefficient~matrix~is~A| +GFd_rFg_r-FH6#%L(c)~may~be~consistent~for~some~values~of~b|+G-FH6#%<(d)~is~alw ays~inconsistent|+GFf\nC(-FH6#%[oIf~the~determinant~of~a~matrix~A~is~not~0,~the n~the~homogeneous|+G-FH6#%Zsystem~of~linear~equations~whose~coefficient~matrix~ is~A|+GFd_rFg_rFj_rF]`rF__nC(-FH6#%^oIf~the~determinant~of~a~matrix~A~is~not~0, ~then~the~nonhomogeneous|+GFd`rFd_r-FH6#%E(b)~has~a~unique~solution~for~any~b|+ GFg`rFj`rFaanC(-FH6#%gnIf~B~is~an~nxn~matrix~obtained~from~a~matrix~A~by~replac ing|+G-FH6#%Othe~i-th~row~of~A~with~2*row1+row2~of~A,~then|+G-FH6#%5(a)~det(B)= 2*det(A)|+G-FH6#%4(b)~det(B)=-det(A)|+G-FH6#%6(c)~det(B)=-2*det(A)|+G-FH6#%3(d) ~det(B)=det(A)|+GF_cnC(-FH6#%_oIf~the~second~row~of~a~5x5~matrix~A~is~a~multipl e~of~the~third~row,|+G-FH6#%Rthen~the~determinant~of~the~matrix~A~is~equal~to|+ G-FH6#%((a)~~0|+GFcfqFfeq-FH6#%((d)~~5|+GFdbpC)-FH6#%[oA~matrix~B~is~obtained~f rom~a~6x6~matrix~A~by~interchanging~the|+G-FH6#%]othird~row~and~the~fifth~row~o f~the~matrix~A.~If~det(B)~is~5,~then|+G-FH6#%Mthe~determinant~of~the~matrix~A~i s~equal~to|+G-FH6#%)(a)~-30|+G-FH6#%((b)~-5|+G-FH6#%)(c)~~30|+GFhcrFfcpC)-FH6#% ]oA~matrix~B~is~obtained~from~a~4x4~matrix~A~by~adding~twice~of~the|+G-FH6#%]ot hird~row~to~the~fourth~row~of~the~matrix~A.~If~det(B)~is~4,~then|+GFbdr-FH6#%(( a)~~8|+G-FH6#%((b)~-8|+G-FH6#%((c)~-4|+G-FH6#%((d)~~4|+GFedpC(-FH6#%joThe~diago nal~elements~of~a~5x5~upper~triangular~matrix~A~are:~-1,2,0,4,5.~Then|+G-FH6#%E the~determinant~of~the~matrix~A~is:|+G-FH6#%)(a)~~40|+G-FH6#%((b)~~0|+G-FH6#%(( c)-40|+GFiaoF^epC(-FH6#%jnGiven~two~4x4~matrices~A~and~B.~If~the~det(A)=5~and~d et(AB)=10|+G-FH6#%Ithen~the~determinant~of~the~matrix~B~is|+G-FH6#%((a)~~5|+G-F H6#%((b)~-2|+G-FH6#%((c)~-5|+G-FH6#%((d)~~2|+GFdepC(-FH6#%]oGiven~two~5x5~matri ces~A~and~B.~If~the~det(A)=2~and~the~det(AB)=0|+G-FH6#%6then~the~matrix~B~is|+G F_dp-FH6#%D(b)~row~equivalent~to~the~identity|+GFafoFiaoF`enFcenF$F$F$, I/questionset2_6:F\jlF^jlF$F$C&@$F[[mC$F][mFTF`[m@AFd[mC(F`bqFcbqFfbqFibqF\cqF_ cqFb^mC)Fhdq>F\wF\eqFi`mF`eqFceqFfeqFieqFc_mC(F]fqF`fqFfbqFd^r-FH6#%((c)~-1|+GF ifqFjamC)Fegn>F\w-F\\m6#7%F^\oF_\oF^hnFi`mF_hnFbhnF`ioFhhnFfcmC2F]gq>F\wFagqFi` m-FH6#%iothe~cofactor~of~each~entry~is~equal~to~the~entry~itself,~then~the~matr ix~A~is|+G>Fj]oFjgq>F\^oF`hq>F9Fagq>F]tFhhqF\iq-FA6#/Fj]o-F_]mF`iqFaiq-FA6#/F\^ o-F_]mFeiqFfiq-FA6#/F9-F_]mFjiqF[jq-FA6#/F]t-F_]mF_jqFbemC'-FH6#%doIf~A~is~an~i nvertible~2x2~matrix,~then~the~determinant~of~the~adjoint(A)|+GF[\rF^\rFa\rFd\r FcfmC,-FH6#%;The~adjoint~of~the~matrix|+G-FA6#/F\w-F_]m6#Fc]o-FH6#%3(a)~is~the~ matrix|+G-FA6#/%$AdjGFd\s-FH6#%3(b)~is~the~matrix|+G-FA6#/F\]s-F_]m6#-F\\m6#7%F \hnFain7%F=F=F_s-FH6#%3(c)~is~the~matrix|+G-FA6#/F\]s-F_]m6#-F\\m6#7%F\hn7%F=F_ sF=F^hn-FH6#%3(d)~is~the~matrix|+G-FA6#/F\]s-F_]m6#F]^oF[imC,F^\s-FA6#/F\w-F_]m 6#-F\\m6#7%7%FbxF=FWFainF^hnFf\sFi\sF]]s-FA6#/F\]s-F_]m6#-F\\m6#7%F^ipFain7%F=F =FbxFi]s-FA6#/F\]s-F_]m6#-F\\m6#7%F`^o7%F=FbxF=F_`sFe^sFi\sF]jmC--FH6#%4If~A~is ~the~matrix|+GF^_s-FH6#%MThe~product~of~the~matrix~A~and~its~adjoint|+GFf\s-FA6 #/F\]s-F_]m6#-F\\m6#7%7%FbxF=F=Fh`sF_`sF]]s-FA6#/F\]s-F_]m6#Fg^oFi]s-FA6#/F\]s- F_]m6#-F\\m6#7%F^ipFh`sF_`sFe^s-FA6#/F\]sFa_sF\[nC,-FH6#%jnThe~determinant~of~a ~matrix~A~is~2~and~its~adjoint~is~given~by|+G-FA6#/F\]s-F_]m6#-F\\m6#7%F\hnFain F`^o-FH6#%4(a)The~matrix~A~is|+G-FA6#/F\wFcas-FH6#%4(b)The~matrix~A~is|+G-FA6#/ F\w-F_]m6#-F\\m6#7%F\hnFh`sF`^o-FH6#%4(c)The~matrix~A~is|+G-FA6#/F\w-F_]m6#-F\\ m6#7%F`^oFh`sF\hn-FH6#%4(d)The~matrix~A~is|+G-FA6#/F\wF`csFg[nC,-FH6#%DThe~adjo int~of~the~diagonal~matrix|+G-FA6#/F\w-F_]m6#-F]^p6%FbxFgwFj\mFf\s-FA6#-F_]m6#- F]^p6%FfdpFg\nFj\mF]]s-FA6#-F_]m6#-F]^p6%Fg\nFfdpFj\mFi]s-FA6#-F_]m6#-F]^p6%Fj\ mFg\nFfdpFe^s-FA6#-F_]m6#-F]^p6%FfdpFj\mFg\nFf\nC,Fhes-FA6#/F\w-F_]m6#-F]^p6%Fb xF=Fj\mFf\s-FA6#F^hsF]]s-FA6#-F_]m6#-F]^p6%F=Fg\nF=Fi]s-FA6#-F_]m6#-F]^p6%FbxFg \nFj\mFe^s-FA6#-F_]m6#-F]^p6%Fj\mFg\nFbxF__nC(-FH6#%epThe~relations~satisfied~b y~a,b,c,and~d~for~which~the~matrix~A~is~equal~to~its~adjoint~are|+G-FA6#/F\w-F_ ]m6#Fagq-FH6#%6(a)~a~=~d~and~b~=~-c|+G-FH6#%7(b)~a~=~-d~and~b~=~-c|+G-FH6#%6(c) ~a~=~-d~and~b~=~c|+G-FH6#%9(d)~a~=~d~and~b~=~c~=~0|+GFaanC'-FH6#%dpIf~the~deter minant~of~a~5x5~matrix~A~is~2,~then~the~determinant~of~the~adjoint~matrix~is|+G -FH6#%'(a)~8|+G-FH6#%((b)~16|+GF\cq-FH6#%'(d)~2|+GF_cnC'-FH6#%7The~adjoint~matr ix~is|+G-FH6#%B(a)~equal~to~the~cofactor~matrix|+G-FH6#%;(b)~equal~to~the~matri x~A|+G-FH6#%S(c)~equal~to~the~transpose~of~the~cofactor~matrix|+G-FH6#%L(d)~equ al~to~the~transpose~of~the~matrix~A|+GF`enFcenF$F$F$, I/questionset2_7:F\jlF^jlF$F$C&@$F[[mC$F][mFT-FH6#%?|+************************* ***|+G@AFd[mC'-FH6#%jnThe~LU-decomposition~of~a~matrix~A~consists~of~the~produc t~of~|+G-FH6#%C(a)~two~lower~triangular~matrices|+G-FH6#%C(b)~two~upper~triangu lar~matrices|+G-FH6#%N(c)~an~upper~and~a~lower~triangular~matrices|+G-FH6#%Q(d) ~an~upper~triangular~and~a~diagonal~matrices|+GFb^mC.-FH6#%4Given~the~matrix~A| +GF\]n-FH6#%XThe~upper~triangular~matrix~in~its~LU-decomposition~is|+G-FA6#/%"U G-F_]m6#-F\\m6#7$Fd]n7$F=Fbx-FH6#%Z(a)~The~lower~triangualr~matrix~in~this~deco mposition~is|+G-FA6#/%"LG-F_]m6#-F\\m6#7$F`coFe]n-FH6#%Z(b)~The~lower~triangual r~matrix~in~this~decomposition~is|+G-FA6#/Fh_tFc^q-FH6#%Z(c)~The~lower~triangua lr~matrix~in~this~decomposition~is|+G-FA6#/Fh_t-F_]m6#-F\\m6#7$F`coF_bn-FH6#%Z( d)~The~lower~triangualr~matrix~in~this~decomposition~is|+G-FA6#/Fh_t-F_]m6#-F\\ m6#7$F`coFg_qFc_mC.Fb^tF\]n-FH6#%XThe~lower~triangular~matrix~in~its~LU-decompo sition~is|+GFe_t-FH6#%Z(a)~The~upper~triangualr~matrix~in~this~decomposition~is |+G-FA6#/F[_tFj]q-FH6#%Z(b)~The~upper~triangualr~matrix~in~this~decomposition~i s|+GFh^t-FH6#%Z(c)~The~upper~triangualr~matrix~in~this~decomposition~is|+G-FA6# /F[_t-F_]m6#-F\\m6#7$Fd]n7$F=Fh`m-FH6#%Z(d)~The~upper~triangualr~matrix~in~this ~decomposition~is|+G-FA6#/F[_t-F_]m6#-F\\m6#7$F\hmFa_tFjamC.-FH6#%.The~matrix~A |+G-FA6#/F\w-F_]m6#-F\\m6#7%F\hnFain7%F_sF=FW-FH6#%cpcan~be~converted~into~an~u pper~triangular~by~premultiplying~it~by~the~elementary~matrix|+G-FA6#/Fjbo-F_]m 6#-F\\m6#7%F^\oFainF\hn-FH6#%fo(a)~The~lower~triangualr~matrix~in~the~LU-decomp osition~of~the~matrix~A~is|+G-FA6#/Fh_t-F_]m6#-F\\m6#7%F^\oFainFjdt-FH6#%fo(b)~ The~lower~triangualr~matrix~in~the~LU-decomposition~of~the~matrix~A~is|+G-FA6#/ Fh_tFaet-FH6#%fo(c)~The~lower~triangualr~matrix~in~the~LU-decomposition~of~the~ matrix~A~is|+G-FA6#/Fh_t-F_]m6#-F\\m6#7%F^\oFain7%F_sF=F_s-FH6#%fo(d)~The~lower ~triangualr~matrix~in~the~LU-decomposition~of~the~matrix~A~is|+G-FA6#/Fh_t-F_]m 6#-F\\m6#7%F^\oFainF`^oFfcmC.F_dt-FA6#/F\w-F_]m6#-F\\m6#7%F`^oFainF`^oF[et-FA6# /FjboF\ft-FH6#%bo(a)~The~lower~triangualr~matrix~in~the~LU-decomposition~of~mat rix~A~is|+GFiet-FH6#%bo(b)~The~lower~triangualr~matrix~in~the~LU-decomposition~ of~matrix~A~is|+GFdft-FH6#%bo(c)~The~lower~triangualr~matrix~in~the~LU-decompos ition~of~matrix~A~is|+GFjft-FH6#%bo(d)~The~lower~triangualr~matrix~in~the~LU-de composition~of~matrix~A~is|+GFfgtFbemC/F_dt-FA6#/F\w-F_]m6#-F\\m6#7%7%FWFWFW7%F WFbxFW7%FWFWFbx-FH6#%jpcan~be~converted~into~an~upper~triangular~by~premultiply ing~it~by~the~elementary~matrices~E2E1|+G-FA6#/%#E1G-F_]m6#-F\\m6#7%F^\o7%F_sFW F=F^hn-FA6#/%#E2GF\ftFjhtFietF]itFdftF`itFjftFcit-FA6#/Fh_t-F_]m6#-F\\m6#7%F^\o FjinF\hnFcfmC)-FH6#%GThe~LU~decomposition~of~a~matrix~A~is|+GFa`tFabt-FH6#%@(a) ~The~matrix~A~is~invertible|+G-FH6#%>(b)~The~matrix~A~is~singular|+G-FH6#%>(c)~ The~matrix~A~is~diagonal|+G-FH6#%@(d)~The~matrix~A~is~triangular|+GF[imC)F\\uFa `tFabt-FH6#%7(a)~Then~det(A)~is~-1|+G-FH6#%6(b)~Then~det(A)~is~1|+G-FH6#%6(c)~T hen~det(A)~is~2|+G-FH6#%7(d)~Then~det(A)~is~-2|+GF]jmC-Fb^t-FA6#/F\wF^`q-FH6#%h nThe~lower~triangular~matrix~in~its~LU-decomposition,~if~any,|+G-FH6#%B(a)~is~t he~inverse~of~the~matrix|+G-FA6#/Fe\mFi_t-FH6#%B(b)~is~the~inverse~of~the~matri x|+G-FA6#/Fe\m-F_]m6#-F\\m6#7$Fd]n7$FWFh`m-FH6#%B(c)~is~the~inverse~of~the~matr ix|+G-FA6#/Fe\m-F_]m6#-F\\m6#7$Fd]n7$FWFg\n-FH6#%B(d)~is~the~inverse~of~the~mat rix|+G-FA6#/Fe\mFictF\[nC-Fb^t-FA6#/F\w-F_]m6#-F\\m6#7$7$F[bmFW7$FgwFbx-FH6#%UT he~lower~triangular~matrix~in~its~LU-decomposition|+GF_^u-FA6#/Fe\m-F_]m6#-F\\m 6#7$7$F[bmF=7$Fd`mF[bmFe^u-FA6#/Fe\m-F_]m6#-F\\m6#7$7$F`_pF=7$#FgwF[bmFWFa_u-FA 6#/Fe\m-F_]m6#-F\\m6#7$7$F\`pF=7$#Fd`mF[bmFWF]`u-FA6#/Fe\mFg`uFg[nC-Fb^t-FA6#/F \w-F_]m6#-F\\m6#7$Fd]nF]auF^auF_^u-FA6#/Fe\m-F_]m6#-F\\m6#7$Fe]nFacoFe^u-FA6#/F e\mFhcuFa_u-FA6#/Fe\mF[coF]`u-FA6#/Fe\m-F_]m6#-F\\m6#7$F`co7$Fd`mFWFf\nC'-FH6#% OWhich~of~the~following~statements~is~correct?|+G-FH6#%P(a)~Every~matrix~has~a~ unique~LU-decomposition|+G-FH6#%O(b)~Some~matrices~have~many~LU-decompositions| +G-FH6#%gn(c)~The~lower~triangular~matrix,~L,~in~A~=~LU~is~invertible|+G-FH6#%e n(d)~The~lower~triangular~matrix,~L,~in~A~=~LU~is~singular|+GF__nC'-FH6#%MIf~a~ matrix~A~has~an~LU-decomposition,~then|+G-FH6#%Z(a)~L~is~the~inverse~of~a~produ ct~of~elementary~matrices|+G-FH6#%:(b)~L~is~always~singular|+G-FH6#%<(c)~L~is~a ~diagonal~matrix|+GFiaoFaanC,-FH6#%FGiven~the~system~of~linear~equations|+G-FA6 #/,(%#x1GFW%#x2GFh`m%#x3GFWFW-FA6#/,(FiguF_sFjguFWF[huF_sFbx-FA6#/,(FiguFWFjguF bxF[huF_sFgcm-FH6#%YThe~solution~y~=~[y1,y2,y3]~of~the~problem~Ly~=~[1,2,5]|+G- FH6#%_owhere~L~is~the~lower~triangular~matrix~in~the~LU-decomposition~of~A|+G-F H6#%2(a)~y~=~[-1,3,1]|+G-FH6#%2(b)~y~=~[1,3,-1]|+G-FH6#%1(c)~y~=~[1,2,5]|+G-FH6 #%2(d)~y~=~[-1,2,5]|+GF_cnC+Fbgu-FA6#/,&FiguFWFjguFh`mFio-FA6#/,&FiguF_sFjguFW" #@-FH6#%VThe~solution~y~=~[y1,y2]~of~the~problem~Ly~=~[15,21]|+GFghu-FH6#%1(a)~ y~=~[36,15]|+G-FH6#%1(b)~y~=~[15,21]|+G-FH6#%3(c)~y~=~[-57,-36]|+G-FH6#%1(d)~y~ =~[15,36]|+GF`enFcenF$F$F$, I*lasterror%Zunable~to~read~`d://publish//ilat4//testxt//lsysquiz.txt`GF$, I'maquiz=F$F%E\[l(%(quiz2_3G:F$F)F$F$C6F4>F9F:>FF?F=-FA6#%JQuiz~on~Singular ~and~Nonsingular~MatricesG-FA6#%F-------------------------------------GFGFKFNFQ FT-FH6#%inType~;~and~press~Enter~to~continue;~or~exit;~to~quit~the~test|+G>FfnF gn@$FjnC7>F^oFW>F`oFW>FboFco>FeoFa[l>F[p66F`pF_pF_pF^pF^pF^pF^pF_pF_pF]pF_pF`pF _pF_pF_pF]pF`pF_pF`pF`p>FbpFW?(F$FWFWF$FdpC$>FhpFj[l@$F`qC%-%,question2_3GFgq>F boF[r>FbpF`r>F?FWFTFbrFgnFT@$FfrC$FhrFTF[sF`sFcsFT@$FjnC$>F]tFfs@+F_tFatFdtFftF itF[uF^uF`uFcuFT@$FguC%Fh]l>F[pFgn@%F_vC&FevFhvFT?(F\wFWFWF]wF^wC&F`w-FadoFjw>F ]xFgn@%F_xFcxC$FgxFjxF\yFT-FH6#%fn~Thank~you~for~using~the~automated~testing~sy stem~of~ILAT.|+GFTFe^l-FH6#%GTo~take~another~quiz,~type~quiz2_3();|+GF[_l>F[pFg nF$F\z%(quiz2_2G:F$F)F$F$C6F4>F9F:>FF?F=-FA6#%D~~~Quiz~on~Special~Type~of~M atricesGF]\vFGFKFNFQFTF`\v>FfnFgn@$FjnC7>F^oFW>F`oFW>FboFco>FeoFfo>F[p61F]pF^pF ]pF_pF]pF]pF_pF`pF`pF`pF_pF`pF_pF`pF`p>FbpFW?(F$FWFWF$FdpC$>FhpFip@$F`qC%-%,que stion2_2GFgq>FboF[r>FbpF`r>F?FWFTFbrFgnFT@$FfrC$FhrFTF[sF`sFcsFT@$FjnC$>F]tFfs@ +F_tFatFdtFftFitF[uF^uF`uFcuFT@$FguC%Fh]l>F[pFgn@%F_vC&FevFhvFT?(F\wFWFWF]wF^wC &F`w-FfenFjw>F]xFgn@%F_xFcxC$FgxFjxF\yFTFh^vFTFe^l-FH6#%GTo~take~another~quiz,~ type~quiz2_2();|+GF[_l>F[pFgnF$F\z%(quiz2_1G:F$F)F$F$C6F4>F9F:>FF?F=-FA6#%7 Quiz~on~Matrix~AlgebraGF]\vFGFKFNFQFT-FH6#%hnType~;~and~press~Enter~to~continue ~or~exit;~to~quit~the~test|+G>FfnFgn@$FjnC7>F^oFW>F`oFW>FboFco>FeoFfo>F[p61F`pF ]pF^pF^pF`pF`pF`pF^pF_pF_pF`pF`pF`pF`pF_p>FbpFW?(F$FWFWF$FdpC$>FhpFip@$F`qC%-%, question2_1GFgq>FboF[r>FbpF`r>F?FWFTFbrFgnFT@$FfrC$FhrFTF[sF`sFcsFT@$FjnC$>F]tF fs@+F_tFatFdtFftFitF[uF^uF`uFcuFT@$FguC%Fh]l>F[pFgn@%F_vC&FevFhvFT?(F\wFWFWF]wF ^wC&F`w-FjilFjw>F]xFgn@%F_xFcxC$FgxFjxF\yFTFh^vFTFe^l-FH6#%GTo~take~another~qui z,~type~quiz2_1();|+GF[_l>F[pFgnF$F\z%(quiz2_7G:F$F)F$F$C6F4>F9F:>FF?F=-FA6 #%9Quiz~on~LU-DecompositionGF]\vFGFKFNFQFTF`\v>FfnFgn@$FjnC7>F^oFW>F`oFW>FboFco >FeoFfo>F[p61F_pF]pF`pF]pF`pF^pF]pF`pF]pF_pF^pF_pF]pF`pF^p>FbpFW?(F$FWFWF$FdpC$ >FhpFip@$F`qC%-%,question2_7GFgq>FboF[r>FbpF`r>F?FWFTFd\lFgnFT@$FfrC$FhrFTF[sF` sFcsFT@$FjnC$>F]tFfs@+F_tFatFdtFftFitF[uF^uF`uFcuFT@$FguC%Fh]l>F[pFgn@%F_vC&Fev FhvFT?(F\wFWFWF]wF^wC&F`w-Fh\tFjw>F]xFgn@%F_xFcxC$FgxFjxF\yFTFh^vFTFe^l-FH6#%GT o~take~another~quiz,~type~quiz2_7();|+GF[_l>F[pFgnF$F\z%(quiz2_6G:F$F)F$F$C6F4> F9F:>FF?F=-FA6#%FfnFgn@$Fjn C7>F^oFW>F`oFW>FboFco>FeoFfo>F[p61F^pF`pF`pF]pF]pF_pF^pF_pF]pF`pF]pF`pF^pF`pF_p >FbpFW?(F$FWFWF$FdpC$>FhpFip@$F`qC%-%,question2_6GFgq>FboF[r>FbpF`r>F?FWFTFbrFg nFT@$FfrC$FhrFTF[sF`sFcsFT@$FjnC$>F]tFfs@+F_tFatFdtFftFitF[uF^uF`uFcuFT@$FguC%F h]l>F[pFgn@%F_vC&FevFhvFT?(F\wFWFWF]wF^wC&F`w-F^irFjw>F]xFgn@%F_xFcxC$FgxFjxF\y FTFh^vFTFe^l-FH6#%GTo~take~another~quiz,~type~quiz2_6();|+GF[_l>F[pFgnF$F\z%(qu iz2_4G:F$F)F$F$C6F4>F9F:>FF?F=-FA6#%9Quiz~on~Matrix~InversionGF]\vFGFKFNFQF TF]gl>FfnFgn@$FjnC7>F^oFW>F`oFW>FboFco>FeoFfo>F[p61F_pF^pF`pF_pF`pF^pF^pF`pF_pF _pF_pF`pF`pF_pF]p>FbpFW?(F$FWFWF$FdpC$>FhpFip@$F`qC%-%,question2_4GFgq>FboF[r>F bpF`r>F?FWFTFbrFgnFT@$FfrC$FhrFTF[sF`sFcsFT@$FjnC$>F]tFfs@+F_tFatFdtFftFitF[uF^ uF`uFcuFT@$FguC%Fh]l>F[pFgn@%F_vC&FevFhvFT?(F\wFWFWF]wF^wC&F`w-F\fpFjw>F]xFgn@% F_xFcxC$FgxFjxF\yFTFh^vFTFe^l-FH6#%GTo~take~another~quiz,~type~quiz2_4();|+GF[_ l>F[pFgnF$F\z%(quiz2_5G:F$F)F$F$C6F4>F9F:>FF?F=-FA6#%5Quiz~on~DeterminantsG F]\vFGFKFNFQFTF]gl>FfnFgn@$FjnC7>F^oFW>F`oFW>FboFco>FeoFa[l>F[p66F^pF`pF`pF_pF] pF]pF_pF^pF_pF]pF_pF_pF`pF^pF]pF`pF^pF`pF^pF_p>FbpFW?(F$FWFWF$FdpC$>FhpFj[l@$F` qC%-%,question2_5GFgq>FboF[r>FbpF`r>F?FWFTFbrFgnFT@$FfrC$FhrFTF[sF`sFcsFT@$FjnC $>F]tFfs@+F_tFatFdtFftFitF[uF^uF`uFcuFT@$FguC%Fh]l>F[pFgn@%F_vC&FevFhvFT?(F\wFW FWF]wF^wC&F`w-FiaqFjw>F]xFgn@%F_xFcxC$FgxFjxF\yFTFh^vFTFe^l-FH6#%GTo~take~anoth er~quiz,~type~quiz2_5();|+GF[_l>F[pFgnF$F\zF$, I&blank:6#%"tG6#F*F$F$?(F\wFWFWFe[mF^w-FA6#%"~GF$F$F$, I,question2_1:6$%(quesnumGF/6#%'count1GF$F$C)-Fjil6#Fe[m-FH6#%XEnter~your~answe r~below~OR~type~exit;~to~quit~the~test|+G>F`o,&F`oFWFWFWFT>FfnFgn>F\wFW@$Fjn@%/ 9%FfnC$-FA6#%FVERY~GOOD.~This~is~the~correct~answerG>F<,&FFfnFgn@%FicwC$F\dw>F&Feo6#F^o6%Fe[mFjcwFbp>F^o,&F^oFWFWFW>F\w,&F\wFWFWFWF$6'F`oF(c)~does ~not~have~a~solution|+G-FH6#%;(d)~has~a~unique~solution|+GFbemC'-FH6#%UThe~grap hs~of~x~-~6y~+~z~=~0~and~2x~-~12y~+~2z~=~1~|+G-FH6#%:(a)~intersect~at~a~point|+ G-FH6#%9(b)~intersect~at~a~line|+G-FH6#%9(c)~parallel~planes~~~~|+G-FH6#%8(d)~o verlapping~planes|+GFcfmC)-FH6#%_oThe~system~of~linear~equations~x+2y-z=0~and~x -y+z=1~has~a~solution.|+G-FH6#%^oSuppose~we~add~the~new~equation~3x+3y-z=k~to~t hese~equations.~Then|+G-FH6#%9the~resulting~system~is|+G-FH6#%V(a)~has~infinite ly~many~solutions~for~any~value~of~k|+G-FH6#%N(b)~has~a~unique~solution~for~any ~value~of~k|+G-FH6#%fn(c)~has~an~empty~solution~set~regardless~of~the~value~of~ k|+G-FH6#%J(d)~has~infinitely~many~solutions~if~k=1|+GF[imC)-FH6#%_oIf~the~solu tion~set~of~a~non-homogeneous~system~of~linear~equations|+G-FH6#%]ois~given~by~ x0+tx1,~then~a~solution~of~the~associated~homogeneous|+G-FH6#%,system~is:|+G-FH 6#%+(a)~x0-x1|+G-FH6#%((b)~x0|+G-FH6#%((c)~x1|+G-FH6#%+(d)~x0+x1|+GF]jmC'-FH6#% IThe~two~equations~x+2y-z=0~and~x-y+z=1~|+G-FH6#%T(a)~intersect~at~a~line~passi ng~through~the~origin|+G-FH6#%en(b)~intersect~at~a~line~passing~through~the~poi nt~(0,1,2)|+G-FH6#%A(c)~intersect~at~a~single~point|+G-FH6#%en(d)~intersect~at~ a~line~passing~through~the~point~(1,1,0)|+GF\[nC'-FH6#%KThe~two~equations~2x+2y -2z=6~and~x+y-z=1~|+GFa`x-FH6#%en(b)~intersect~at~a~line~passing~through~the~po int~(0,2,1)|+G-FH6#%D(c)~represents~two~parallel~planes|+G-FH6#%fn(d)~intersect ~at~a~line~passing~through~the~point~(0,1,-2)|+GFg[nC(-FH6#%_oIf~three~planes~h ave~a~line~in~common,~then~the~system~of~equations|+G-FH6#%=represented~by~thes e~planes|+GF_gw-FH6#%C(b)~has~infinitely~many~solutions|+G-FH6#%5(c)~has~no~sol ution|+GFiaoFf\nC(-FH6#%YIf~three~planes~have~a~point~in~common,~then~the~syste m|+G-FH6#%Mof~linear~equations~defined~by~these~planes|+G-FH6#%C(a)~has~infinit ely~many~solutions|+G-FH6#%;(b)~has~a~unique~solution|+GFdbx-FH6#%G(d)~has~only ~three~distinct~solutions|+GF__nC(-FH6#%_oIf~three~planes~intersect~such~that~a ny~two~have~a~distinct~line~in|+G-FH6#%^ocommon,~the~the~system~of~linear~equat ions~defined~by~these~planes|+G-FH6#%A(a)~must~have~a~unique~solution|+G-FH6#%I (b)~must~have~infinitely~many~solutions|+G-FH6#%:(c)~must~be~inconsistent|+G-FH 6#%H(d)~none~of~the~above~will~always~hold|+GFaanC(-FH6#%]oIf~three~lines~inter sect~such~that~any~two~have~distinct~point~in|+G-FH6#%^ocommon,~then~the~system ~of~linear~equations~defined~by~these~lines|+GF^dxFadxFddxFgdxF_cnC'-FH6#%aoWhi ch~one~of~the~following~statements~is~correct?~The~solution~set~of|+G-FH6#%\o(a )~every~nonhomogeneous~system~of~linear~equations~is~non-empty|+G-FH6#%in(b)~ev ery~homogeneous~system~of~linear~equations~is~non-empty|+G-FH6#%en(c)~every~hom ogeneous~system~of~linear~equations~is~empty|+G-FH6#%hn(d)~Every~nonhomogeneous ~system~of~linear~equations~is~empty|+G-FA6#%@Add~More~Questions~to~the~test!G- FH6#%1|+**************|+GF$F$F$, I,question2_2:FibwF[cwF$F$C)-FfenF_cwF`cw>F`oFdcwFT>FfnFgn>F\wFW@$Fjn@%FicwC$F\ dw>FFfnFgn@%FicwC$F\dw>FF`ewFbew>F^oFdew>F\wFfe wF$FgewF$, I/questionset1_2:F\jlFjewF$F$C&@$F[[mC$F][mFTFafw@AFd[mC'F\hw-FH6#%:(a)~is~alwa ys~consistent|+G-FH6#%<(b)~is~always~inconsistent|+G-FH6#%I(c)~does~only~have~t he~trivial~solution|+G-FH6#%I(d)~does~only~have~nontrivial~solutions|+GFb^mC'F\ iwFchxFfhxFihxFiaoFc_mC)-FH6#%]oThe~system~of~linear~equations~x+2y-z=0~and~x-y +z=1~is~consistent|+G-FH6#%inSuppose~we~add~the~new~equation~3x+3y-z=k~to~these ~equations.|+G-FH6#%;Then~the~resulting~system|+G-FH6#%_o(a)~is~consistent~with ~infinitely~many~solutions~for~any~value~of~k|+G-FH6#%gn(b)~is~consistent~with~ a~unique~solution~for~any~value~of~k|+G-FH6#%Y(c)~is~always~inconsistent~regard less~of~the~value~of~k|+G-FH6#%Y(d)~is~consistent~with~infinitely~many~solution s~if~k=1|+GFjamC)-FH6#%^oIf~the~solution~set~of~a~nonhomogeneous~system~of~line ar~equations|+G-FH6#%^ois~given~by~x0+t.x1,~then~a~solution~of~the~associated~h omogeneous|+GF^_xFa_xFd_xFg_xFj_xFfcmC)-FH6#%\oA~nonhomogeneous~system~of~linea r~equations~has~infinitely~many~|+G-FH6#%\osolutions~given~by~x~=~(1,2,-4)~+~t* (-1,4,5)~where~t~is~any~real|+G-FH6#%\oparameter.Then~the~solution~of~the~assoc iated~homogeneous~system|+G-FH6#%4(a)~is~t*(-1,4,5)~|+G-FH6#%<(b)~is~(1-t,2+4*t ,-4-5*t)~|+G-FH6#%B(c)~is~only~the~trivial~solution|+G-FH6#%1(d)~is~(0,6,1)~|+G FbemC(-FH6#%]oA~nonhomogeneous~system~of~linear~equations~has~a~unique~solution |+G-FH6#%hngiven~by~x=(-1,0,3,5).Then~the~associated~homogeneous~system|+G-FH6# %G(a)~has~t*(-1,0,3,5)~as~its~solution~|+GFbgwFehw-FH6#%O(d)~has~a~solution~tha t~cannot~be~determined~|+GFcfmC)-FH6#%]oThe~system~of~linear~equations~x+2y-z=1 ~and~x-y+z=0~is~consistent|+GFdixFgix-FH6#%L(a)~is~inconsistent~if~k~is~not~equ al~to~2|+GF]jx-FH6#%_o(c)~is~consistent~with~infinitely~many~solutions~for~any~ value~of~k|+G-FH6#%H(d)~is~inconsistent~if~k~is~equal~to~2|+GF[imC(-FH6#%@~~The ~system~~2x~-~2y~-~2z~=~4|+G-FH6#%@~~~~~~~~~~~~~~~x~-~~y~-~~z~=~2|+G-FH6#%5(a)~ is~inconsistent|+GFh[x-FH6#%3(c)~is~consistent|+GF^\xF]jmC'F^im-FH6#%]o(a)~Ever y~nonhomogeneous~system~of~linear~equations~is~consistent|+G-FH6#%jn(b)~Every~h omogeneous~system~of~linear~equations~is~consistent|+G-FH6#%\o(c)~Every~homogen eous~system~of~linear~equations~is~inconsistent|+G-FH6#%_o(d)~Every~nonhomogene ous~system~of~linear~equations~is~inconsistent|+GF\[nC*Fijw-FH6#%6~~~x~-~3*y~-~ k*z~=~0|+G-FH6#%6-2*x~+~2*y~+~~~z~=~1|+G-FH6#%6~~-x~-~~y~-~~~~z~=~2|+G-FH6#%E(a )~is~consistent~for~k~equals~to~2|+G-FH6#%H(b)~is~consistent~for~k~not~equal~to ~2|+G-FH6#%Q(c)~is~consistent~regardless~of~the~values~of~k|+GF]`rFg[nC(Fh`p-FH 6#%VEvery~nonhomogeneous~system~of~linear~equations~with|+G-FH6#%V(a)~more~equa tions~than~unknowns~may~be~inconsistent|+G-FH6#%jn(b)~less~equations~than~unkno wns~has~infinitely~many~solutions|+G-FH6#%X(c)~less~equations~than~unknowns~has ~a~unique~solution|+G-FH6#%T(d)~more~equations~than~unknowns~may~be~consistent| +GFf\nC(-FH6#%_oIf~a~homogeneous~system~of~n~equations~with~n~unknowns~has~only ~the|+G-FH6#%gozero~solution,~then~the~solution~of~the~associated~nonhomogeneou s~system~is|+G-FH6#%1(a)~is~infinite|+G-FH6#%4(b)~is~a~singleton|+G-FH6#%.(c)~i s~empty|+G-FH6#%2(d)~is~undefined|+GF__nC(-FH6#%fnIf~a~nonhomogeneous~system~ha s~a~unique~solution,~then~the|+G-FH6#%Vsolution~set~of~the~associated~homogeneo us~system~is|+GFaby-FH6#%.(b)~is~empty|+G-FH6#%4(c)~is~a~singleton|+GFjbyFaanC( -FH6#%hnIf~a~homogeneous~system~has~more~than~one~solution,~then~the|+G-FH6#%Ys olution~set~of~the~associated~nonhomogeneous~system~is|+GFaby-FH6#%/(b)~is~uniq ue|+GFgby-FH6#%:(d)~cannot~be~determined|+GF_cnC*Fijw-FH6#%4~~~x~+~y~-~k*z~=~0| +G-FH6#%4~~~x~-~y~+~~~z~=~1|+G-FH6#%4~~~~~~~(1-k)*z~=~2|+G-FH6#%E(a)~is~consist ent~for~k~equals~to~1|+G-FH6#%H(b)~is~consistent~for~k~not~equal~to~1|+GFg`yFj` rFafx-FH6#%7|+********************|+GF$F$F$, I,question2_3:FibwF[cwF$F$C)-FadoF_cwF`cw>F`oFdcwFT>FfnFgn>F\wFW@$Fjn@%FicwC$F\ dw>FFfnFgn@%FicwC$F\dw>FF`ewFbew>F^oFdew>F\wFfe wF$FgewF$, I/questionset1_3:F\jlFjewF$F$C&@$F[[mF][mFafw@AFd[mC)-FH6#%-The~matrix~|+G>F\w- F\\m6#7%F^\oF]hnF^hnFi`mF[]p-FH6#%I(b)~is~an~example~of~an~identity~matrix|+G-F H6#%=(c)~is~an~elementary~matrix|+GFiaoFb^mC*Fdan>F\w-F\\m6#7&7&FWF=F=,&F``mFWF c`mFh`m7&F=FWFgwF=7&F=F=FWF=Fb[oFi`m-FH6#%enThen~the~matrix~is~an~example~of~an ~elementary~matrix~if~|+G-FH6#%5(a)~a1,a2~arbitrary|+G-FH6#%/(b)~a1-2*a2=0|+G-F H6#%+(c)~a2=~0|+G-FH6#%+(d)~a1=~0|+GFc_mC*Fdan>F\wFdhyFi`m-FH6#%inThen~the~matr ix~is~not~an~example~of~an~elementary~matrix~if~|+G-FH6#%?(a)~a1~=~0~and~a2~is~ not~zero|+GFaiy-FH6#%7(c)~a1~=~0~and~a2~=~0|+GFiaoFjamC,-FH6#%DThe~system~of~li near~equations~A:~|+G-FA6#/F^dmFW-FA6#/,&Fa\mFbxFb\mFd`mFW-FH6#%@is~equivalent~ to~the~system~B:|+GFijy-FA6#/,&Fa\mFgcmFb\m!"'F[bm-FH6#%in(a)~since~B~is~obtain ed~from~A~by~an~elementary~row~operation|+G-FH6#%V(b)~since~the~two~systems~hav e~the~same~solution~set|+G-FH6#%fn(c)~since~the~two~systems~represent~two~inter secting~lines|+G-FH6#%N(d)~answer(a)~and~answer(b)~are~both~correct|+GFfcmC,Ffj y-FA6#/,(Fa\mFWFb\mFWFc\mFWFW-FA6#/,(Fa\mFWFb\mF_sFc\mFWFWF`[zFe\z-FA6#/,&Fa\mF WFc\mFWFW-FH6#%gn(a)~since~B~is~obtained~from~A~by~elementary~row~operations|+G F[\z-FH6#%gn(c)~since~the~two~systems~represent~two~intersecting~planes|+GFa\zF bemC,FfjyFijy-FA6#/,&Fa\mFbxFb\mFgwFW-FH6#%Jcan~never~be~equivalent~to~the~syst em~B:|+GFijy-FA6#/F[^zFgcm-FH6#%`o(a)~since~B~cannot~be~obtained~from~A~by~an~e lementary~row~operation|+G-FH6#%en(b)~since~the~two~systems~have~the~different~ solution~set|+GF^\zFa\zFcfmC(-FH6#%]oIf~a~homogeneous~system~of~n~equations~in~ n~unknowns~has~only~the|+GF^byFabyFdbyFgbyFjbyF[imC(-FH6#%`oIf~a~nonhomogeneous ~system~of~linear~equations~has~a~unique~solution|+G-FH6#%inthen~the~solution~s et~of~the~associated~homogeneous~system~is|+GFabyFdcyFgcyFjbyF]jmC(-FH6#%boIf~a ~homogeneous~system~of~linear~equations~has~more~than~one~solution|+G-FH6#%\oth en~the~solution~set~of~the~associated~nonhomogeneous~system~is|+GFabyFadyFgbyFd dyF\[nC*FijwFhdyF[eyF^eyFaeyFdeyFg`yFj`rFg[nC*FijwFhdyF[ey-FH6#%4~~~~~~~(1-k)*z ~=~1|+G-FH6#%N(a)~has~a~unique~solution~if~k~is~equal~to~1|+G-FH6#%R(b)~has~a~u nique~solution~if~k~is~not~equal~to~1|+G-FH6#%Y(c)~has~a~unique~solution~regard less~of~the~values~of~k|+G-FH6#%F(d)~can~never~have~a~unique~solution|+GFf\nC*F ijwFhdyF[ey-FH6#%4~2*x~+~(1-k)*z~=~1|+G-FH6#%V(a)~has~infinitely~many~solutions ~if~k~is~equal~to~1|+G-FH6#%Z(b)~has~infinitely~many~solutions~if~k~is~not~equa l~to~1|+G-FH6#%[o(c)~has~infinitely~many~solutions~regardless~of~the~values~of~ k|+G-FH6#%N(d)~can~never~have~infinitely~many~solutions|+GF__nC*FijwFhdy-FH6#%4 ~~~x~-~y~+~2*z~=~1|+G-FH6#%4~2*x~+~(2-k)*z~=~3|+G-FH6#%W(a)~has~infinitely~many ~solutions~if~k~is~equal~to~-2|+G-FH6#%Z(b)~has~infinitely~many~solutions~if~k~ is~not~equal~to~2|+GFeazF]`rFaanC*FijwFhdyF\bzF_bz-FH6#%<(a)~is~always~inconsis tent|+G-FH6#%J(b)~is~consistent~if~k~is~not~equal~to~2|+G-FH6#%F(c)~is~consiste nt~if~k~is~equal~to~2|+G-FH6#%<(d)~is~inconsistent~if~k=2|+GF_cnC(-FH6#%BThe~so lution~set~of~the~equation|+G-FH6#%2~~~x~+~y~-~z~=~0|+G-FH6#%@(a)~includes~one~ free~variable|+G-FH6#%A(b)~includes~two~free~variables|+G-FH6#%C(c)~includes~th ree~free~variables|+GFiaoFafxFgeyF$F$F$, I,question2_4:FibwF[cwF$F$C)-F\fpF_cwF`cw>F`oFdcwFT>FfnFgn>F\wFW@$Fjn@%FicwC$F\ dw>FFfnFgn@%FicwC$F\dw>FF`ewFbew>F^oFdew>F\wFfe wF$FgewF$, I/questionset1_4:F\jlFjewF$F$C&@$F[[mF][mFafw@AFd[mC0-FH6#%7The~augmented~matri x~|+G>F\w-F\\m6#7%7%FbxFWFWF]hn7%F=F=F=Fi`m-FH6#%<(a)~represents~the~system:|+G -FA6#/,&Fa\mFbxFb\mFWFW-FA6#/Fb\mFbx-FH6#%<(b)~represents~the~system:|+GF\gz-FA 6#/F`\mFbx-FH6#%<(c)~represents~the~system:|+GF\gzFfgz-FA6#/F=F=FiaoFb^mC*-FH6# %DThe~augmented~matrix~of~the~system|+GF\[xF_[xFb[x-FH6#%0(a)~4x3~matrix|+G-FH6 #%0(b)~3x4~matrix|+G-FH6#%0(c)~3x3~matrix|+G-FH6#%0(d)~4x4~matrix|+GFc_mC*F`hz- FH6#%6~2x~-~2y~-~2z~-w~=~0|+GF_[x-FH6#%8~~x~-~~y~-~~3z~-5w~=~2|+GFchzFfhz-FH6#% 0(c)~3x5~matrix|+GF\izFjamC)F`hzF\[xF_[x-FH6#%0(a)~2x3~matrix|+G-FH6#%0(b)~3x2~ matrix|+GFihz-FH6#%0(d)~2x4~matrix|+GFfcmC*-FH6#%WThe~(3,4)~entry~in~the~augmen ted~matrix~of~the~system|+GF`izF_[xFciz-FH6#%+(a)~is~-3|+G-FH6#%+(b)~is~-5|+GF[ anF^anFbemC+-FH6#%KGiven~the~augmented~matrix~of~a~system~of|+G-FH6#%2linear~eq uations|+G>F\wF\`mFi`m-FH6#%SThen~the~system~has~infinitely~many~solutions~if~| +GF^amFaam-FH6#%-(c)~a2=3*a1|+GFgamFcfmC*F^[[lFa[[l>F\wFdfn-FAFew-FH6#%in(a)~Th e~system~has~infinitely~many~solutions~for~any~a1,a2,a3|+G-FH6#%X(b)~The~system ~has~a~unique~solution~for~any~a1,a2,~a3|+GF^gnFagnF[imC)FaixFe]x-FH6#%6the~res ulting~system|+GFjixF]jxF`jxFcjxF]jmC+-FH6#%UThe~augmented~matrix~associated~wi th~a~given~system|+G-FH6#%=of~homogeneous~equations~is|+G>F\w-F\\m6#7%7'FWF_sFW F=F=7'F=FWFWFWF=7'F=F=FWF_sF=Fi`m-FH6#%?In~solving~the~system~we~have|+G-FH6#%9 (a)~four~free~variables|+G-FH6#%7(b)~one~free~variable|+G-FH6#%:(c)~three~free~ variables|+G-FH6#%8(d)~two~free~variables|+GF\[nC+Fi\[lF\][l>F\w-F\\m6#7%7(FWF_ sFWFWFWF=7(F=F=FWFWFWF=7(F=F=F=FWF_sF=Fi`mFf][lFi][lF\^[lF_^[lFb^[lFg[nC+Fi\[lF \][l>F\w-F\\m6#7$Fj^[l7(F=F=F=F=F=F=Fi`mFf][lFi][lF\^[lF_^[lFb^[lFf\nC+Fi\[lF\] [l>F\w-F\\m6#7%Fj^[l7(F=F=F=F=FWF=Fh_[lFi`mFf][lFi][lF\^[lF_^[lFb^[lF__nC*Fijw- FH6#%6~~~x~-~3*y~-~2*z~=~0|+GF[`yF^`y-FH6#%3(a)~is~consistent|+GFh[x-FH6#%5(c)~ is~inconsistent|+GF^\xFaanC(F^^yFa^yFd^yFh[xFg^yF^\xF_cnC*Fi\n>F\w-F\\m6#7%7'F= FWFWFWFW7'F=F=F=FWFW7'F=F=F=F=F=Fi`m-FH6#%Sis~the~coefficient~matrix~of~a~homog eneous~system|+G-FH6#%G(a)~This~system~has~one~free~variable|+G-FH6#%H(b)~This~ system~has~two~free~variables|+G-FH6#%J(c)~This~system~has~three~free~variables |+G-FH6#%I(d)~This~system~has~four~free~variables|+GFafxFgeyF$F$F$, I,question2_5:FibwF[cwF$F$C)-FiaqF_cwF`cw>F`oFdcwFT>FfnFgn>F\wFW@$Fjn@%FicwC$F\ dw>FFfnFgn@%FicwC$F\dw>FF`ewFbew>F^oFdew>F\ wFfewF$FgewF$, I,question2_6:FibwF[cwF$F$C)-F^irF_cwF`cw>F`oFdcwFT>FfnFgn>F\wFW@$Fjn@%FicwC$F\ dw>FFfnFgn@%FicwC$F\dw>FF`ewFbew>F^oFdew>F\wFfe wF$FgewF$, I/questionset1_5:F\jlFjewF$F$C&@$F[[mC$F][mFTFafw@KFd[mC)Fegy>F\w-F\\m6#7%7%F=F WFWF^hnFhfzF]\[l-FH6#%=(a)~is~in~row~echelon~form~|+G-FH6#%@(b)~is~in~reduced~e chelon~form|+GFfipFiaoFb^mC(-FH6#%[oThe~gauss~elimination~algorithm~reduces~the ~augmented~matrix~of|+G-FH6#%Jan~associated~system~of~linear~equations|+G-FH6#% D(a)~to~a~matrix~in~an~echelon~form|+G-FH6#%C(b)~to~an~upper~triangular~matrix| +G-FH6#%B(c)~to~a~lower~triangular~matrix|+G-FH6#%=(d)~to~a~tridiagonal~matrix| +GFc_mC+-FH6#%hnGiven~the~echelon~form~of~an~augmented~matrix~of~a~system~of|+G Fa[[l>F\wF\`mF]\[lFe[[lF^amFaamFh[[lFgamFjamC*F\g[lFa[[l>F\wFdfnF]\[lF^\[lFa\[l F^gnFagnFfcmC)FaixFe]xFh]x-FH6#%\o(a)~consistent~with~infinitely~many~solutions ~for~any~value~of~k|+G-FH6#%Z(b)~consistent~with~a~unique~solution~for~any~valu e~of~k|+G-FH6#%V(c)~always~inconsistent~regardless~of~the~value~of~k|+G-FH6#%V( d)~consistent~with~infinitely~many~solutions~if~k=1|+GFbemC+-FH6#%\oThe~echelon ~form~of~the~augmented~matrix~associated~with~a~given|+G-FH6#%Dsystem~of~homoge neous~equations~is|+G>F\wF`][lFi`mFf][lFi][lF\^[lF_^[lFb^[lFcfmC+F`h[lFch[l>F\w Fg^[lFi`mFf][lFi][lF\^[lF_^[lFb^[lF[imC+F`h[lFch[l>F\wF__[lFi`mFf][lFi][lF\^[lF _^[lFb^[lF]jmC+F`h[lFch[l>F\wFe_[lFi`mFf][lFi][lF\^[lF_^[lFb^[lF\[nC)-FH6#%6The ~matrix~given~by:|+G>F\w-F\\m6#7%F^hnFae[lFhfzFi`m-FH6#%<(a)~is~in~row~echelon~ form|+G-FH6#%D(b)~is~in~reduced~row~echelon~form|+G-FH6#%@(c)~is~not~in~row~ech elon~form|+G-FH6#%jn(d)~is~in~row~echelon~form~but~not~in~reduced~row~echelon~f orm|+GFg[nC'-FH6#%`oThe~backsubstitution~algorithm~should~be~implemented~after~ computing|+G-FH6#%\o(a)~the~inverse~of~the~augmented~matrix~associated~with~a~s ystem|+G-FH6#%^o(b)~the~transpose~of~the~augmented~matrix~associated~with~a~sys tem|+G-FH6#%`o(c)~the~row~echelon~of~the~augmented~matrix~associated~with~a~sys tem|+G-FH6#%[o(d)~the~square~of~the~augmented~matrix~associated~with~a~system|+ GFf\nC*Fi\n>F\wF^e[lFi`m-FH6#%>is~an~example~of~a~matrix~in|+G-FH6#%7(a)~row~ec helon~form~|+G-FH6#%>(b)~reduced~row~echelon~form|+GFfipFiaoF__nC*Fi\n>F\wFf`[l Fi`m-FH6#%\ois~the~coefficient~matrix~of~a~homogeneous~system~in~row~echelon|+G F_a[lFba[lFea[lFha[lFaanC.Fi\n>F\w-F\\m6#7%7'F=FWFWF_sFWFj`[lF[a[lFi`mF^\\l>Fj] o-Fgo6#7'Figu,&F[huF_s%#x5GFh`mF[hu,$F\]\lF_sF\]\l>F\^o-Fgo6#7'F=F[]\lF[huF]]\l F\]\l>F9-Fgo6#7'F=,$F[huF_sF[huF]]\lF\]\l>F]t-Fgo6#7'FiguFf]\lF[huF]]\lF\]\l-FA 6%%4(a)~The~solution~isGF\^o%F~where~x1,~x3,~x5~are~free~variables|+G-FA6%%4(b) ~The~solution~isGFj]oF^^\l-FA6%%4(c)~The~solution~isGF9F^^\l-FA6%%4(d)~The~solu tion~isGF]tF^^\lF_cnC.Fi\n>F\wFc\\lFi`m-FH6#%]ois~the~augmented~matrix~of~a~non homogeneous~system~in~row~echelon|+G>Fj]o-Fgo6#7&Figu,&FbxFWF[huF_sF[huFW>F\^o- Fgo6#7&F=Fa_\lF[hu%#x4G>F9-Fgo6#7&F=Fa_\lF[huFW>F]t-Fgo6#7&Figu,&FbxFWF[huFWF[h uFW-FA6%F]^\lF\^o%E~where~x3~and~x4~are~free~variables|+G-FA6%Fa^\lF]t%J~where~ x1,~x3,~and~x4~are~free~variables|+G-FA6%Fd^\lF9%>~where~x3~is~a~free~variable| +G-FA6%Fg^\lFj]o%E~where~x1~and~x3~are~free~variables|+GFdbpC+F\g[lFa[[l>F\wFdf nFi`m-FH6#%KThen~the~system~has~a~unique~solution~if~|+GF^am-FH6#%0(b)~a1-a2-a3 =1|+GFh[[lFgamFfcpC+F\g[lFa[[l>F\w-F\\m6#7%7&FWFgcmF,Fj\m7&F=FWFbxF_s7&F=F=,&F, FWF_sFWFWFi`mF^a\l-FH6#%)(a)~k=1|+G-FH6#%9(b)~k~is~not~equal~to~1|+G-FH6#%:(c)~ k~is~any~real~number|+GFiaoFedpC+F\g[lFa[[l>F\w-F\\m6#7%Fia\lFja\l7&F=F=F\b\lF= Fi`mFe[[l-FH6#%:(a)~k~is~any~real~number|+GF`b\l-FH6#%)(c)~k=1|+GFiaoF^epC(Ffcz FiczF\dzF_dzFbdzFiaoFdepC(Ffcz-FH6#%5~~~x~+~y~-~z+~w~=~0|+GF\dzF_dzFbdzFiaoFafx FgeyF$F$F$, I,question2_7:FibwF[cwF$F$C)-Fh\tF_cwF`cw>F`oFdcwFT>FfnFgn>F\wFW@$Fjn@%FicwC$F\ dw>FFfnFgn@%FicwC$F\dw>FF`ewFbew>F^oFdew>F\wFfe wF$FgewF$, I,question1_1:FibwF[cwF$F$C)-F`blF_cw-FH6#%fn<~Enter~your~answer~below~OR~type~ exit;~to~quit~the~test~>|+G>F`oFdcwFT>FfnFgn>F\wFW@$Fjn@%FicwC%F\dwFT>FFfnFgn@%FicwC%F\dwFT>FF`ewFbew>F^oFdew>F\wFfewF$FgewF$, I,question1_2:FibwF[cwF$F$C)-FiwF_cwF`cw>F`oFdcwFT>FfnFgn>F\wFW@$Fjn@%FicwC$-FA 6#%GVERY~GOOD.~You~gave~the~correct~answerG>FFfnFgn@%FicwC$F^ g\l>FF`ewFbew>F^oFdew>F\wFfewF$FgewF$, I,question1_3:FibwF[cwF$F$C)-FielF_cwF`e\l>F`oFdcwFT>FfnFgn>F\wFW@$Fjn@%FicwC$- FA6#%GVERY~GOOD.~This~is~~the~correct~answerG>FFfnFgn@%FicwC$ F\dw>FF`ewFbew>F^oFdew>F\wFfewF$FgewF$, I,question1_4:FibwF[cwF$F$C)-FbilF_cwF`cw>F`oFdcwFT>FfnFgn>F\wFW@$Fjn@%FicwC$F\ dw>FFfnFgn@%FicwC$F\dw>FF`ewFbew>F^oFdew>F\wFfe wF$FgewF$, I,question1_5:FibwF[cwF$F$C)-Fa^lF_cwF`e\l>F`oFdcwFT>FfnFgn>F\wFW@$Fjn@%FicwC$F fh\l>FFfnFgn@%FicwC$F\dw>FF`ewFbew>F^oFdew>F\wF fewF$FgewF$