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H6#%I~~To~take~another~quiz,~type~quiz1_2();|+G-FH6#%=~~To~exit~press~;~and~ent
er|+G>F[pFgnF$6*F`oF<FeoF^oFbpF?Ffn%'finishG%(quiz1_5G:F$F)F$F$C6F4>F9F:>F<F=>F
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uiz1_1G:F$F)F$F$C6F4>F9F:>F<F=>F?F=-FA6#%T~~~~~~~~~~~~~~~~~Quiz~on~Examples~of~
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pFgnF$F\z%(quiz1_3G:F$F)F$F$C6F4>F9F:>F<F=>F?F=-FA6#%S~~~~~~~~~~~~~~~~~Quiz~on~
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F=-FA6#%hn~~~~~~~~~~~~~~Quiz~on~Matrix~Representation~of~Linear~SystemsG-FA6#%j
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UMBER~%d|+GFbp-FH6#%H|+*************************************|+G@A/FbjlFWC)-FH6#
%@The~system~of~linear~equations|+G-FH6#%.~2x~+~2y~=~5|+G-FH6#%.~~x~+~~y~=~2|+G
-FH6#%;(a)~has~a~unique~solution|+G-FH6#%5(b)~has~no~solution|+G-FH6#%C(c)~has~
infinitely~many~solutions|+G-FH6#%6(d)~has~two~solutionsG/FbjlFbxC'-FH6#%MAny~h
omogeneous~system~of~linear~equations:|+G-FH6#%?(a)~has~at~least~one~solution|+
G-FH6#%?(b)~has~an~empty~solution~set|+G-FH6#%C(c)~has~only~the~trivial~solutio
n|+G-FH6#%C(d)~has~only~nontrivial~solutions|+G/FbjlFgwC'-FH6#%PAny~nonhomogene
ous~system~of~linear~equations:|+G-FH6#%B(a)~has~a~non-empty~solution~set|+GFi]
mF\^m-FH6#%7(d)~none~of~the~above|+G/Fbjl""%C*-FH6#%CThe~solution~to~the~linear
~system|+G-FH6#%.~2x~+~2y~=~0|+G-FH6#%.~~x~-~~y~=~0|+G-FH6#%$is|+G-FH6#%,(a)~[2
,-2]|+G-FH6#%+(b)~[0,0]|+G-FH6#%,(c)~[-1,1]|+GFj^m/Fbjl""&C*-FH6#%,The~system|+
G-FH6#%3~2x~-~2y~-~2z~=~0|+G-FH6#%3-2x~+~2y~+~~z~=~1|+G-FH6#%3~~x~-~~y~-~~z~=~2
|+G-FH6#%4(a)~has~a~solution|+G-FH6#%6(b)~is~a~homogeneous|+G-FH6#%>(c)~does~no
t~have~a~solution|+G-FH6#%;(d)~has~a~unique~solution|+G/Fbjl""'C'-FH6#%UThe~gra
phs~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)~
overlapping~planes|+G/Fbjl""(C)-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~these~equations.~Then|+G-FH6#%9the~resulting~system~is|+G-FH6#%V(a)~has~inf
initely~many~solutions~for~any~value~of~k|+G-FH6#%N(b)~has~a~unique~solution~fo
r~any~value~of~k|+G-FH6#%fn(c)~has~an~empty~solution~set~regardless~of~the~valu
e~of~k|+G-FH6#%J(d)~has~infinitely~many~solutions~if~k=1|+G/Fbjl"")C)-FH6#%_oIf
~the~solution~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-FH6#%+(a)~x0-x1|+G-FH6#%((b)~x0|+G-FH6#%((c)~x1|+G-FH6#%+(d)~x0+x1|+G/Fb
jl""*C'-FH6#%IThe~two~equations~x+2y-z=0~and~x-y+z=1~|+G-FH6#%T(a)~intersect~at
~a~line~passing~through~the~origin|+G-FH6#%en(b)~intersect~at~a~line~passing~th
rough~the~point~(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)|+G/FbjlF:C'-FH6#%KThe~two
~equations~2x+2y-2z=6~and~x+y-z=1~|+GFhfm-FH6#%en(b)~intersect~at~a~line~passin
g~through~the~point~(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)|+G/Fbjl"#6C(-FH6#
%_oIf~three~planes~have~a~line~in~common,~then~the~system~of~equations|+G-FH6#%
=represented~by~these~planes|+GFe\m-FH6#%C(b)~has~infinitely~many~solutions|+G-
FH6#%5(c)~has~no~solution|+GFj^m/Fbjl"#7C(-FH6#%YIf~three~planes~have~a~point~i
n~common,~then~the~system|+G-FH6#%Mof~linear~equations~defined~by~these~planes|
+G-FH6#%C(a)~has~infinitely~many~solutions|+G-FH6#%;(b)~has~a~unique~solution|+
GF^im-FH6#%G(d)~has~only~three~distinct~solutions|+G/Fbjl"#8C(-FH6#%_oIf~three~
planes~intersect~such~that~any~two~have~a~distinct~line~in|+G-FH6#%^ocommon,~th
e~the~system~of~linear~equations~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-FH6#%H(d)~none~of~the~above~will~always~hold|+G/Fbjl"
#9C(-FH6#%]oIf~three~lines~intersect~such~that~any~two~have~distinct~point~in|+
G-FH6#%^ocommon,~then~the~system~of~linear~equations~defined~by~these~lines|+GF
\[nF_[nFb[nFe[n/FbjlFioC'-FH6#%aoWhich~one~of~the~following~statements~is~corre
ct?~The~solution~set~of|+G-FH6#%\o(a)~every~nonhomogeneous~system~of~linear~equ
ations~is~non-empty|+G-FH6#%in(b)~every~homogeneous~system~of~linear~equations~
is~non-empty|+G-FH6#%en(c)~every~homogeneous~system~of~linear~equations~is~empt
y|+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/questionset1_2:FhjlFjjlF$F$C&@$Fb[mC$Fc[mFTFf[m@AFj[mC'Fc]m-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|+GFa]mC'Fd
^mF_^nFb^nFe^nFj^mFb^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|+GF]_mC)-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|+GFcemFfemFiemF\fmF_fmFe`mC)-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
F`bmC(-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~|+GFh\mF\^m-FH6#%O(d)~has~a~solution~tha
t~cannot~be~determined~|+GFbcmC)-FH6#%]oThe~system~of~linear~equations~x+2y-z=1
~and~x-y+z=0~is~consistent|+GF`_nFc_n-FH6#%L(a)~is~inconsistent~if~k~is~not~equ
al~to~2|+GFi_n-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|+GFjdmC(-FH6#%@~~The
~system~~2x~-~2y~-~2z~=~4|+G-FH6#%@~~~~~~~~~~~~~~~x~-~~y~-~~z~=~2|+G-FH6#%5(a)~
is~inconsistent|+GFgam-FH6#%3(c)~is~consistent|+GF]bmFbfmC'-FH6#%SWhich~one~of~
the~following~statements~is~correct?|+G-FH6#%]o(a)~Every~nonhomogeneous~system~
of~linear~equations~is~consistent|+G-FH6#%jn(b)~Every~homogeneous~system~of~lin
ear~equations~is~consistent|+G-FH6#%\o(c)~Every~homogeneous~system~of~linear~eq
uations~is~inconsistent|+G-FH6#%_o(d)~Every~nonhomogeneous~system~of~linear~equ
ations~is~inconsistent|+GFdgmC*Fh`m-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~eq
uals~to~2|+G-FH6#%H(b)~is~consistent~for~k~not~equal~to~2|+G-FH6#%Q(c)~is~consi
stent~regardless~of~the~values~of~k|+G-FH6#%5(d)~is~inconsistent|+GFbhmC(-FH6#%
QWhich~one~of~the~following~statements~is~false?|+G-FH6#%VEvery~nonhomogeneous~
system~of~linear~equations~with|+G-FH6#%V(a)~more~equations~than~unknowns~may~b
e~inconsistent|+G-FH6#%jn(b)~less~equations~than~unknowns~has~infinitely~many~s
olutions|+G-FH6#%X(c)~less~equations~than~unknowns~has~a~unique~solution|+G-FH6
#%T(d)~more~equations~than~unknowns~may~be~consistent|+GFaimC(-FH6#%_oIf~a~homo
geneous~system~of~n~equations~with~n~unknowns~has~only~the|+G-FH6#%gozero~solut
ion,~then~the~solution~of~the~associated~nonhomogeneous~system~is|+G-FH6#%1(a)~
is~infinite|+G-FH6#%4(b)~is~a~singleton|+G-FH6#%.(c)~is~empty|+G-FH6#%2(d)~is~u
ndefined|+GFcjmC(-FH6#%fnIf~a~nonhomogeneous~system~has~a~unique~solution,~then
~the|+G-FH6#%Vsolution~set~of~the~associated~homogeneous~system~is|+GFfhn-FH6#%
.(b)~is~empty|+G-FH6#%4(c)~is~a~singleton|+GF_inFh[nC(-FH6#%hnIf~a~homogeneous~
system~has~more~than~one~solution,~then~the|+G-FH6#%Ysolution~set~of~the~associ
ated~nonhomogeneous~system~is|+GFfhn-FH6#%/(b)~is~unique|+GF\in-FH6#%:(d)~canno
t~be~determined|+GFa\nC*Fh`m-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~consistent~for~k~equals~to~1|+G-
FH6#%H(b)~is~consistent~for~k~not~equal~to~1|+GFffn-FH6#%<(d)~is~always~inconsi
stent|+GFb]n-FH6#%7|+********************|+GF$F$F$,
I/questionset1_3:FhjlFjjlF$F$C&@$Fb[mFc[mFf[m@AFj[mC)-FH6#%-The~matrix~|+G>F\w-
%'matrixG6#7%7%FWF=F=7%F=FWFbx7%F=F=FW-FA6#/F\w-%&evalmGFew-FH6#%A(a)~is~not~an
~elementary~matrix|+G-FH6#%I(b)~is~an~example~of~an~identity~matrix|+G-FH6#%=(c
)~is~an~elementary~matrix|+GFj^mFa]mC*-FH6#%2Given~the~matrix|+G>F\w-F]]o6#7&7&
FWF=F=,&%#a1GFW%#a2G!"#7&F=FWFgwF=7&F=F=FWF=7&F=F=F=FWFc]o-FH6#%enThen~the~matr
ix~is~an~example~of~an~elementary~matrix~if~|+G-FH6#%5(a)~a1,a2~arbitrary|+G-FH
6#%/(b)~a1-2*a2=0|+G-FH6#%+(c)~a2=~0|+G-FH6#%+(d)~a1=~0|+GFb^mC*Fb^o>F\wFf^oFc]
o-FH6#%inThen~the~matrix~is~not~an~example~of~an~elementary~matrix~if~|+G-FH6#%
?(a)~a1~=~0~and~a2~is~not~zero|+GFg_o-FH6#%7(c)~a1~=~0~and~a2~=~0|+GFj^mF]_mC,-
FH6#%DThe~system~of~linear~equations~A:~|+G-FA6#/,&%"xGFW%"yGF_sFW-FA6#/,&FcaoF
bxFdao!"$FW-FH6#%@is~equivalent~to~the~system~B:|+GF_ao-FA6#/,&FcaoFf`mFdao!"'F
^_m-FH6#%in(a)~since~B~is~obtained~from~A~by~an~elementary~row~operation|+G-FH6
#%V(b)~since~the~two~systems~have~the~same~solution~set|+G-FH6#%fn(c)~since~the
~two~systems~represent~two~intersecting~lines|+G-FH6#%N(d)~answer(a)~and~answer
(b)~are~both~correct|+GFe`mC,F\ao-FA6#/,(FcaoFWFdaoFW%"zGFWFW-FA6#/,(FcaoFWFdao
F_sFccoFWFWFjaoF_co-FA6#/,&FcaoFWFccoFWFW-FH6#%gn(a)~since~B~is~obtained~from~A
~by~elementary~row~operations|+GFebo-FH6#%gn(c)~since~the~two~systems~represent
~two~intersecting~planes|+GF[coF`bmC,F\aoF_ao-FA6#/,&FcaoFbxFdaoFgwFW-FH6#%Jcan
~never~be~equivalent~to~the~system~B:|+GF_ao-FA6#/FfdoFf`m-FH6#%`o(a)~since~B~c
annot~be~obtained~from~A~by~an~elementary~row~operation|+G-FH6#%en(b)~since~the
~two~systems~have~the~different~solution~set|+GFhboF[coFbcmC(-FH6#%]oIf~a~homog
eneous~system~of~n~equations~in~n~unknowns~has~only~the|+GFchnFfhnFihnF\inF_inF
jdmC(-FH6#%`oIf~a~nonhomogeneous~system~of~linear~equations~has~a~unique~soluti
on|+G-FH6#%inthen~the~solution~set~of~the~associated~homogeneous~system~is|+GFf
hnFiinF\jnF_inFbfmC(-FH6#%boIf~a~homogeneous~system~of~linear~equations~has~mor
e~than~one~solution|+G-FH6#%\othen~the~solution~set~of~the~associated~nonhomoge
neous~system~is|+GFfhnFfjnF\inFijnFdgmC*Fh`mF][oF`[oFc[oFf[oFi[oFffnF\\oFbhmC*F
h`mF][oF`[o-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~unique~solution~if~k~is~not~equal~to~1|+G-FH6#%Y
(c)~has~a~unique~solution~regardless~of~the~values~of~k|+G-FH6#%F(d)~can~never~
have~a~unique~solution|+GFaimC*Fh`mF][oF`[o-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~equal~to~1|+G-FH6#%[o(c)~has~infinitely~many~soluti
ons~regardless~of~the~values~of~k|+G-FH6#%N(d)~can~never~have~infinitely~many~s
olutions|+GFcjmC*Fh`mF][o-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|+GF`hoFifnFh[nC*Fh`mF][oFghoF
jho-FH6#%<(a)~is~always~inconsistent|+G-FH6#%J(b)~is~consistent~if~k~is~not~equ
al~to~2|+G-FH6#%F(c)~is~consistent~if~k~is~equal~to~2|+G-FH6#%<(d)~is~inconsist
ent~if~k=2|+GFa\nC(-FH6#%BThe~solution~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~va
riables|+G-FH6#%C(c)~includes~three~free~variables|+GFj^mFb]nF_\oF$F$F$,
I/questionset1_4:FhjlFjjlF$F$C&@$Fb[mFc[mFf[m@AFj[mC0-FH6#%7The~augmented~matri
x~|+G>F\w-F]]o6#7%7%FbxFWFWFa]o7%F=F=F=Fc]o-FH6#%<(a)~represents~the~system:|+G
-FA6#/,&FcaoFbxFdaoFWFW-FA6#/FdaoFbx-FH6#%<(b)~represents~the~system:|+GFb\p-FA
6#/,&FcaoFWFdaoFWFbx-FH6#%<(c)~represents~the~system:|+GFb\pF\]p-FA6#/F=F=Fj^mF
a]mC*-FH6#%DThe~augmented~matrix~of~the~system|+GF[amF^amFaam-FH6#%0(a)~4x3~mat
rix|+G-FH6#%0(b)~3x4~matrix|+G-FH6#%0(c)~3x3~matrix|+G-FH6#%0(d)~4x4~matrix|+GF
b^mC*Fg]p-FH6#%6~2x~-~2y~-~2z~-w~=~0|+GF^am-FH6#%8~~x~-~~y~-~~3z~-5w~=~2|+GFj]p
F]^p-FH6#%0(c)~3x5~matrix|+GFc^pF]_mC)Fg]pF[amF^am-FH6#%0(a)~2x3~matrix|+G-FH6#
%0(b)~3x2~matrix|+GF`^p-FH6#%0(d)~2x4~matrix|+GFe`mC*-FH6#%WThe~(3,4)~entry~in~
the~augmented~matrix~of~the~system|+GFg^pF^amFj^p-FH6#%+(a)~is~-3|+G-FH6#%+(b)~
is~-5|+G-FH6#%*(c)~is~2|+G-FH6#%+(d)~is~-2|+GF`bmC+-FH6#%KGiven~the~augmented~m
atrix~of~a~system~of|+G-FH6#%2linear~equations|+G>F\w-F]]o6#7%7&FWF=FWF[_o7&F=F
WFbx,&F\_oFWF[_oFiao7&F=F=F=,(%#a3GFWF\_oF]_oF[_oFWFc]o-FH6#%SThen~the~system~h
as~infinitely~many~solutions~if~|+G-FH6#%8(a)~a1,a2,a3~arbitrary|+G-FH6#%0(b)~a
1-a2-a3=0|+G-FH6#%-(c)~a2=3*a1|+G-FH6#%0(d)~a3=2*a2-a1|+GFbcmC*F[apF^ap>F\w-F]]
o6#7%FeapFfap7&F=F=FWFiap-FAFew-FH6#%in(a)~The~system~has~infinitely~many~solut
ions~for~any~a1,a2,a3|+G-FH6#%X(b)~The~system~has~a~unique~solution~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|+GFjdmC)F]_nFhcm
-FH6#%6the~resulting~system|+GFf_nFi_nF\`nF_`nFbfmC+-FH6#%UThe~augmented~matrix
~associated~with~a~given~system|+G-FH6#%=of~homogeneous~equations~is|+G>F\w-F]]
o6#7%7'FWF_sFWF=F=7'F=FWFWFWF=7'F=F=FWF_sF=Fc]o-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|+GFdgmC+FbdpFedp>F\w-F]
]o6#7%7(FWF_sFWFWFWF=7(F=F=FWFWFWF=7(F=F=F=FWF_sF=Fc]oF_epFbepFeepFhepF[fpFbhmC
+FbdpFedp>F\w-F]]o6#7$Fcfp7(F=F=F=F=F=F=Fc]oF_epFbepFeepFhepF[fpFaimC+FbdpFedp>
F\w-F]]o6#7%Fcfp7(F=F=F=F=FWF=FagpFc]oF_epFbepFeepFhepF[fpFcjmC*Fh`m-FH6#%6~~~x
~-~3*y~-~2*z~=~0|+GFjenF]fn-FH6#%3(a)~is~consistent|+GFgam-FH6#%5(c)~is~inconsi
stent|+GF]bmFh[nC(FjcnF]dnF`dnFgamFcdnF]bmFa\nC*-FH6#%,The~matrix|+G>F\w-F]]o6#
7%7'F=FWFWFWFW7'F=F=F=FWFW7'F=F=F=F=F=Fc]o-FH6#%Sis~the~coefficient~matrix~of~a
~homogeneous~system|+G-FH6#%G(a)~This~system~has~one~free~variable|+G-FH6#%H(b)
~This~system~has~two~free~variables|+G-FH6#%J(c)~This~system~has~three~free~var
iables|+G-FH6#%I(d)~This~system~has~four~free~variables|+GFb]nF_\oF$F$F$,
I/questionset1_5:FhjlFjjlF$F$C&@$Fb[mC$Fc[mFTFf[m@KFj[mC)Fh\o>F\w-F]]o6#7%7%F=F
WFWFb]oF^\pF`cp-FH6#%=(a)~is~in~row~echelon~form~|+G-FH6#%@(b)~is~in~reduced~ec
helon~form|+G-FH6#%K(c)~is~an~example~of~an~elementary~matrix|+GFj^mFa]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~fo
rm|+G-FH6#%C(b)~to~an~upper~triangular~matrix|+G-FH6#%B(c)~to~a~lower~triangula
r~matrix|+G-FH6#%=(d)~to~a~tridiagonal~matrix|+GFb^mC+-FH6#%hnGiven~the~echelon
~form~of~an~augmented~matrix~of~a~system~of|+GF^ap>F\wFbapF`cpF[bpF^bpFabpFdbpF
gbpF]_mC*F`\qF^ap>F\wF\cpF`cpFacpFdcpFgcpFjcpFe`mC)F]_nFhcmF[dm-FH6#%\o(a)~cons
istent~with~infinitely~many~solutions~for~any~value~of~k|+G-FH6#%Z(b)~consisten
t~with~a~unique~solution~for~any~value~of~k|+G-FH6#%V(c)~always~inconsistent~re
gardless~of~the~value~of~k|+G-FH6#%V(d)~consistent~with~infinitely~many~solutio
ns~if~k=1|+GF`bmC+-FH6#%\oThe~echelon~form~of~the~augmented~matrix~associated~w
ith~a~given|+G-FH6#%Dsystem~of~homogeneous~equations~is|+G>F\wFidpFc]oF_epFbepF
eepFhepF[fpFbcmC+Fd]qFg]q>F\wF`fpFc]oF_epFbepFeepFhepF[fpFjdmC+Fd]qFg]q>F\wFhfp
Fc]oF_epFbepFeepFhepF[fpFbfmC+Fd]qFg]q>F\wF^gpFc]oF_epFbepFeepFhepF[fpFdgmC)-FH
6#%6The~matrix~given~by:|+G>F\w-F]]o6#7%Fb]oFbjpF^\pFc]o-FH6#%<(a)~is~in~row~ec
helon~form|+G-FH6#%D(b)~is~in~reduced~row~echelon~form|+G-FH6#%@(c)~is~not~in~r
ow~echelon~form|+G-FH6#%jn(d)~is~in~row~echelon~form~but~not~in~reduced~row~ech
elon~form|+GFbhmC'-FH6#%`oThe~backsubstitution~algorithm~should~be~implemented~
after~computing|+G-FH6#%\o(a)~the~inverse~of~the~augmented~matrix~associated~wi
th~a~system|+G-FH6#%^o(b)~the~transpose~of~the~augmented~matrix~associated~with
~a~system|+G-FH6#%`o(c)~the~row~echelon~of~the~augmented~matrix~associated~with
~a~system|+G-FH6#%[o(d)~the~square~of~the~augmented~matrix~associated~with~a~sy
stem|+GFaimC*F^hp>F\wF_jpFc]o-FH6#%>is~an~example~of~a~matrix~in|+G-FH6#%7(a)~r
ow~echelon~form~|+G-FH6#%>(b)~reduced~row~echelon~form|+GFijpFj^mFcjmC*F^hp>F\w
FbhpFc]o-FH6#%\ois~the~coefficient~matrix~of~a~homogeneous~system~in~row~echelo
n|+GF[ipF^ipFaipFdipFh[nC.F^hp>F\w-F]]o6#7%7'F=FWFWF_sFWFfhpFghpFc]oFbaq>8%-Fgo
6#7'%#x1G,&%#x3GF_s%#x5GF]_oFbbq,$FcbqF_sFcbq>8&-Fgo6#7'F=FabqFbbqFdbqFcbq>F9-F
go6#7'F=,$FbbqF_sFbbqFdbqFcbq>F]t-Fgo6#7'F`bqF^cqFbbqFdbqFcbq-FA6%%4(a)~The~sol
ution~isGFfbq%F~where~x1,~x3,~x5~are~free~variables|+G-FA6%%4(b)~The~solution~i
sGF\bqFfcq-FA6%%4(c)~The~solution~isGF9Ffcq-FA6%%4(d)~The~solution~isGF]tFfcqFa
\nC.F^hp>F\wFgaqFc]o-FH6#%]ois~the~augmented~matrix~of~a~nonhomogeneous~system~
in~row~echelon|+G>F\bq-Fgo6#7&F`bq,&FbxFWFbbqF_sFbbqFW>Ffbq-Fgo6#7&F=FidqFbbq%#
x4G>F9-Fgo6#7&F=FidqFbbqFW>F]t-Fgo6#7&F`bq,&FbxFWFbbqFWFbbqFW-FA6%FecqFfbq%E~wh
ere~x3~and~x4~are~free~variables|+G-FA6%FicqF]t%J~where~x1,~x3,~and~x4~are~free
~variables|+G-FA6%F\dqF9%>~where~x3~is~a~free~variable|+G-FA6%F_dqF\bq%E~where~
x1~and~x3~are~free~variables|+G/Fbjl"#;C+F`\qF^ap>F\wF\cpFc]o-FH6#%KThen~the~sy
stem~has~a~unique~solution~if~|+GF^bp-FH6#%0(b)~a1-a2-a3=1|+GFdbpFgbp/Fbjl"#<C+
F`\qF^ap>F\w-F]]o6#7%7&FWFf`mF,Fabm7&F=FWFbxF_s7&F=F=,&F,FWF_sFWFWFc]oFhfq-FH6#
%)(a)~k=1|+G-FH6#%9(b)~k~is~not~equal~to~1|+G-FH6#%:(c)~k~is~any~real~number|+G
Fj^m/Fbjl"#=C+F`\qF^ap>F\w-F]]o6#7%FegqFfgq7&F=F=FhgqF=Fc]oF[bp-FH6#%:(a)~k~is~
any~real~number|+GF\hq-FH6#%)(c)~k=1|+GFj^m/Fbjl"#>C(FajoFdjoFgjoFjjoF][pFj^m/F
bjlFc[lC(Fajo-FH6#%5~~~x~+~y~-~z+~w~=~0|+GFgjoFjjoF][pFj^mFb]nF_\oF$F$F$,
I,question1_1:6$%(quesnumGF/6#%'count1GF$F$C)-F`bl6#Fbjl-FH6#%fn<~Enter~your~an
swer~below~OR~type~exit;~to~quit~the~test~>|+G>F`o,&F`oFWFWFWFT>FfnFgn>F\wFW@$F
jn@%/9%FfnC%-FA6#%FVERY~GOOD.~This~is~the~correct~answerGFT>F<,&F<FWFWFW@$/F\wF
WC%-FH6#%6Incorrect.~Try~again|+G>FfnFgn@%FjjqC%F][rFT>F<Fa[rC'-FH6#%5Sorry.~Wr
ong~Again!|+GFT>&Feo6#F^o6%FbjlF[[rFbp>F^o,&F^oFWFWFW>F\w,&F\wFWFWFWF$6'F`oF<Fe
oF^oFfnF$,
I,question1_2:FjiqF\jqF$F$C)-FiwF`jq-FH6#%XEnter~your~answer~below~OR~type~exit
;~to~quit~the~test|+G>F`oFejqFT>FfnFgn>F\wFW@$Fjn@%FjjqC$-FA6#%GVERY~GOOD.~You~
gave~the~correct~answerG>F<Fa[r@$Fc[rC%Fe[r>FfnFgn@%FjjqC$Ff]r>F<Fa[rC'F]\rFT>F
a\rFc\r>F^oFe\r>F\wFg\rF$Fh\rF$,
I,question1_3:FjiqF\jqF$F$C)-FielF`jqFajq>F`oFejqFT>FfnFgn>F\wFW@$Fjn@%FjjqC$-F
A6#%GVERY~GOOD.~This~is~~the~correct~answerG>F<Fa[r@$Fc[rC%Fe[r>FfnFgn@%FjjqC$F
][r>F<Fa[rC'F]\rFT>Fa\rFc\r>F^oFe\r>F\wFg\rF$Fh\rF$,
I,question1_4:FjiqF\jqF$F$C)-FbilF`jqF]]r>F`oFejqFT>FfnFgn>F\wFW@$Fjn@%FjjqC$F]
[r>F<Fa[r@$Fc[rC%Fe[r>FfnFgn@%FjjqC$F][r>F<Fa[rC'F]\rFT>Fa\rFc\r>F^oFe\r>F\wFg\
rF$Fh\rF$,
I,question1_5:FjiqF\jqF$F$C)-Fa^lF`jqFajq>F`oFejqFT>FfnFgn>F\wFW@$Fjn@%FjjqC$F^
_r>F<Fa[r@$Fc[rC%Fe[r>FfnFgn@%FjjqC$F][r>F<Fa[rC'F]\rFT>Fa\rFc\r>F^oFe\r>F\wFg\
rF$Fh\rF$