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D$E@RXx*O ddB`DE@RXăB`DE@@RXăB` C osg DE@ XăG osgCjZ HGL` noc$noc$Z Z - ??????cer$cer$cer$HL cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C1 - 1 D E@OY Y B`9C  deqe , deqe X deqe 1 deqe deqe DE@FFXăGdeqe#* e=88D}HX2 3 LUEquation Version 60Gdeqe#* 9e>88E}HXA LUEquation Version 60Gdeqe#* e?88F}HX`( i , {} j ) LUEquation Version 60Gdeqe#* F@88G}HX a_{ i j } LGEquation Version 60Gdeqe#* FA88H}HXa_{ i j } = i + j - 1 LGEquation Version 60B`'K}C | L, , deqe X 1X i@? ne DE@v#eeXGdeqe#* 9eB88I}HXA LUEquation Version 60G ne)HB` C @onta deqe D E @eeăG onta)G deqe#* eC88J}HX 15 LUEquation Version 60B`DE@3*RXăB`DE@s0RXăB`DE@6RXăB`DE@<RXăB`DE@3CRXăB`DE@sIRXăB` C  osg DE@O XăG osgCjZ HGL` noc$noc$Z Z - ??????cer$cer$cer$HL cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C1 - 2 D E@OY Y B`C DE@ _RXăB`U C  ne, deqe @ deqe ,t, deqe 1@ deqe t. t L, deqe X @? D, EKeo o :Xă>qeeXăG ne)HB`E C onta  b$ ȴ ,@ deqe ,t deqe 1@ deqe t. 4" deqe D  E eRăeeă;5 eeăG onta)G deqe#* eS88}HX 2 times 4LUEquation Version 60G deqe#* eT88}HX`(2,~3)LUEquation Version 60G deqe#* eU88}HX7LUEquation Version 60G deqe#* -eV88}HXm+n+k = 2+4+7 = 13LUEquation Version 60Gdeqe#* Qo I88}HX!pmatrix {1&~2&~3&~4 # 5&~6&~7&~8}LBEquation Version 60Gdeqe#* I eJ88}HXm timesnLUEquation Version 60Gdeqe#* eK88}HX`(2,~3)LUEquation Version 60Gdeqe#* eL88}HXkLUEquation Version 60Gdeqe#* eM88}HX@m+n+kLUEquation Version 60B`dC `$ deqe  a$ deqe j b$ deqe  c$ deqe  d$ deqe DEweeXă?~eeXăGdeqe#* eN88}HX11LUEquation Version 60Gdeqe#* eO88}HX12LUEquation Version 60Gdeqe#* eP88}HX13LUEquation Version 60Gdeqe#* eQ88}HX14LUEquation Version 60Gdeqe#* eR88}HX15LUEquation Version 60B`DE@IRXăB`DE@RXăB`DE@ɑRXăB`DE@ RXăB`DE@IRXăB`DE@RXăB` C  osg DE@ɪ XăG osgCjZ HGL` noc$noc$Z Z - ??????cer$cer$cer$HL cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C1 - 3 D E@OY Y B`e C t(Ȭ, deqe X deqe 1 deqe L t X L, , deqe X deqe X @? lbt DE!FFXă9o o :Xă\00XăGdeqe#* 9eW88}HXALUEquation Version 60Gdeqe#*  eX88}HX`(i,~j)LUEquation Version 60Gdeqe#* FY88}HX a_ijLGEquation Version 60Gdeqe#* to Z88}HXA= pmatrix{a``&b#c```&d}LBEquation Version 60Gdeqe#* e[88}HX@ad-bcLUEquation Version 60G lbt"*j\}HMH j jB`C *8*X G1 D E @xiH 383B`D E @TH kf8kfB` C  deqe D E @P P XPeG deqe#* Q"P ]88}HX@ 5a _{ij} = cases{i+j```&(i >= j)#-a _{ji} ```&(i<j)}LAEquation Version 60H 383B`D E @TH jjB`D E @xiB`]C `$ deqe a$ deqe b$ deqe  c$ deqe d$ deqe DEeeXă<eeXăGdeqe#* e^88}HX-17LUEquation Version 60Gdeqe#* e_88}HX-9LUEquation Version 60Gdeqe#* e`88}HX1LUEquation Version 60Gdeqe#* ea88}HX9LUEquation Version 60Gdeqe#* eb88}HX17LUEquation Version 60B`DE@RXăB`DE@RXăB`DE@RXăB`DE@HRXăB`DE@ RXăB` C  osg DE@ XăG osgCjZ GL` noc$noc$Z Z - ??????cer$cer$cer$L cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C1 -4 D E@OY Y B`4 CL D ̹qXՔ  deqe X deqe X @? lbt DE !eeX+'00XGdeqe#* ec88†}HX x,~yLUEquation Version 60Gdeqe#* eed88Æ}HXxyLUEquation Version 60G lbt"*jeĆ}HMH j jB`C *8* D E @xiH 383B`D E @TH kf8kfB` C @ deqe D E @+ + XPeG deqe#* '+ f88ņ}HX @pmatrix{x+y```&2#8```&5} = pmatrix{3```&2#8```&x ^{2} +y ^{2}}LAEquation Version 60H 383B`D E @TH jjB`D E @xiB`S C` `$ deqe _ a$ deqe G b$ deqe G c$ deqe G d$ deqe DE@e?eeXGdeqe#* eg88Ɔ}HX1LUEquation Version 60Gdeqe#* eh88dž}HX2LUEquation Version 60Gdeqe#* ei88Ȇ}HX3LUEquation Version 60Gdeqe#* ej88Ɇ}HX4LUEquation Version 60Gdeqe#* ek88ʆ}HX5LUEquation Version 60B`DE@"FRXăB`DE@bLRXăB`DE@RRXăB`DE@RXcB`DE@@RXcB`DE@ RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@%RXcB`DE@+RXcB`DE@2RXcB` C@@ǀtǔ.... D'E@@8RXcB`DE@>RXcB`DE@DRXcB`DE@KRXcB`DE@@QRXcB`DE@WRXcB`DE@]RXcB`DE@dRXcB`DE@@jRXcB`DE@pRXcB`DE@vRXcB`DE@}RXcB`DE@@RXcB`DE@RXcB` C@@ǀtǔ.... D'E@RXcB`D'E@RXcB`D'E@@RXcB`D'E@RXcB`D'E@RXcB`D'E@RXcB`D'E@@RXcB`D'E@RXcB`D'E@RXcB`D'E@RXcB`D'E@@RXcB`D'E@RXcB`D'E@RXcB`D'E@RXcB` C@@ǀtǔ.... D'E@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@ RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@%RXcB`DE@,RXcB`DE@@2RXcB` C@@ǀtǔ.... D'E@8RXcB`C  osg osg DE XăRXcG osgCj aGLcer$cer$Xaq w?"?V@ @ 0~1!@ CE ?u <V??7AaH  B `C`PD1 - 2. ,X Ű DE@@@PO aqaqG osgCjaGL` noc$noc$ aqB ??@ @&r?(F%0.}??noc$cer$cer$aL noc$ @&r???\\?c#{???cip$lle$Lcip$ @&r???\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$ @&r???\\?&gc#{?N[Q??+?+???!@H B`C1-1 DE` ` ` ` ` P8! and elsewhere.L cer$ @&r????PE???!@H B`C` DERX@RXO@ @ L cer$ @&r???????!@H B`DE@RXOXXB`C dloc osg DE@'J'J?XăG`dloc1F5G osgCjn'J'bGLcer$cer$ZP,n'J]7%??ٛ?? `~:?@??!A(bH nnB `DE@RXlB `C(1) ,X gH, H, 0 DE@@RXlB `C 1) Typet @ ,| gH, H DE@ RXlB `C@ X @ 1| gH, H DE@RXlB `$ C 2),X 0 : 1 0 deqe DERXl@o o :XlGdeqe#* $,o 7889}HX@ =A= pmatrix { a& b# c& d}```RARROW kA=pmatrix {ka &kb# kc & kdLBEquation Version 60O ZZP,P,B` C  osg DE@LgXăG osgCjaGL` noc$noc$q w@ ??????noc$cer$cer$bL noc$7zq 8 ?K@?? a/@???\\?c#{???cip$lle$Lcip$7ȯ z$t= ?K@?? a/@???\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$I 8!8! \\?@?@?fu@???\\?&gc#{?N[Q??+?+???!@H R B`C2-1 DE@` ` P P8! and elsewhere.L cer$@ @ ??t@?????PE???!@H  B`C` DE@RX O@ @ L cer$XXD,l?@?x@?iҼ@???????!@H )dB`DE@RXaOXXB`!C  P t( Ȭ, deqe deqe , DE@`: : /XGdeqe#* e888=}HX P , {} Q LUEquation Version 60Gdeqe#* y!: 988>}HX ;P+ 2 Q = LEFT ( { 1} atop {2 } { -3} atop {5 } RIGHT ) LCEquation Version 60B`$B}C deqe | ̹q` L, deqe X 1X D"E@l: : /XGdeqe#* ": :88?}HX :P - Q = LEFT ( { 1} atop { -1} { 3} atop {-4 } RIGHT ) LCEquation Version 60Gdeqe#* e;88@}HXP + Q LUEquation Version 60B` C}Ci@? ne DE@xRXG ne)HB` C @onta deqe D E @eeăG onta)G deqe#* e<88A}HX4 LUEquation Version 60B`DE@Z~RXăB`C00 D&E@RXăB`DE@ڊRXăB`DE@RXăB`DE@ZRXăB`DE@RXăB`DE@ڣRXăB`DE@RXăB`DE@ZRXăB`DE@RXăB`DE@ڼRXăB`DE@RXăB`DE@ZRXăB` C  osg DE@gXăG osgCjqbGL` noc$noc$qq 8 ??????noc$cer$cer$bL noc$7zq 8 ?K@?????\\?c#{???cip$lle$Lcip$7ȯ z$t= ?K@?????\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$I 8!8! \\?@?@????\\?&gc#{?N[Q??+?+???!@H  B`C2-2 DE@` `  P8! and elsewhere.L cer$@ @ ??t@?????PE???!@H @ B`C` DE@RX O@ @ L cer$XXD,l?@?x@????????!@H XB`DE@RXPVOXXB`2C@ deqe | ̹qXՔ  deqe X D lX|. ne DE@o o :XăGdeqe#* )!o y88}HXppmatrix{col{ x # 1 }lcol{ 2 # -1 }} pmatrix{col{ 1 # y } lcol{ 3 # -1 }} = pmatrix{col{ 1 # 2 }lcol{ 7 # 4 }} LBEquation Version 60Gdeqe#* ez88}HX`x, y LUEquation Version 60G ne)HB` C @onta deqe D E @eeăG onta)G deqe#* e{88}HX x=3, y=-1 LUEquation Version 60B`C00 D&E@RXăB`DE@RXăB`DE@XRXăB`DE@RXăB`DE@RXăB`DE@RXăB`DE@XRXăB`DE@RXăB`DE@!RXăB`DE@(RXăB`DE@X.RXăB`DE@4RXăB`DE@:RXăB`DE@ARXăB`DE@XGRXăB`DE@RXcB`DE@@RXcB`DE@ RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@%RXcB`DE@+RXcB`DE@2RXcB`DE@@8RXcB`DE@>RXcB`DE@DRXcB`DE@KRXcB`DE@@QRXcB` C@@ǀtǔ.... D'E@WRXcB`DE@]RXcB`DE@dRXcB`DE@@jRXcB`DE@pRXcB`DE@vRXcB`DE@}RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB` C@@ǀtǔ.... D'E@@RXcB`D#E@RXcB`D#E@RXcB`D#E@RXcB`D#E@@RXcB`D#E@RXcB`D#E@RXcB`D#E@RXcB`D#E@@RXcB`D#E@RXcB`D#E@ RXcB`D#E@RXcB`D#E@@RXcB`D#E@RXcB`D#E@%RXcB` C  osg DE@,XcG osgCj,bGLcer$cer$d,Wk??::(@+M????!AbH  ,B `C  ز TIP ƃ̹@.... D$E@RXx*O ddB`DE@/5RXcB`DE@o;RXcB`DE@ARXcB` C  osg DE@ XăG osgCjZ bGL` noc$noc$Z Z - ??????cer$cer$cer$ bL cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C2 - 1 D E@OY Y B`C 8 , DE@XRXB` Q}C` deqe DE@: : /XGdeqe#* Q: D88L}HXA = LEFT ( { 1} atop {0 } { 2} atop {-1 } RIGHT ) , {} B = LEFT ( { -1} atop {1 } { -2} atop {1 } RIGHT ) , {} C = LEFT ( { -1} atop { 2} { -2} atop {1 } RIGHT ) LCEquation Version 60B`$R}C deqe X deqe 1` L, DE@*!eeXGdeqe#* eE88M}HX x , {} y LUEquation Version 60Gdeqe#* eF88N}HXx A + y B = 3 C LUEquation Version 60B`S}C deqe X D lX|. ne DE@'eeXGdeqe#* eG88O}HXx + y LUEquation Version 60G ne)HB` C @onta deqe D E @eeăG onta)G deqe#* eH88P}HX9 LUEquation Version 60B`C DE@.RXăB`DE@4RXăB`DE@$;RXăB`DE@dARXăB`DE@GRXăB`DE@MRXăB` C  osg DE@$T XăG osgCjZ !bGL` noc$noc$Z Z - ??????cer$cer$cer$"bL cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C2 - 2 D E@OY Y B`G C , deqe X deqe | ̹qXՔ  deqe X D lX|. ne DE|co o :Xă%CoeeXăGdeqe#* Ho |88}HX)A = pmatrix{col{ 1 # 3 }lcol{ 1 # 2 }} LBEquation Version 60Gdeqe#* !}88}HXA^3 = xA +y E LWEquation Version 60Gdeqe#* e~88 }HX`x, y LUEquation Version 60G ne)HB` C `onta deqe D E @eeăG onta)G deqe#* e88#}HX x=10, y=3 LUEquation Version 60B`C@ DE@vRXăB`DE@@|RXăB`DE@RXăB`DE@RXăB`DE@RXăB`DE@@RXăB`DE@RXăB`DE@RXăB` C  osg DE@ XăG osgCjZ #bGL` noc$noc$Z Z - ??????cer$cer$cer$$bL cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C2 - 3 D E@OY Y B`'C , deqe | L, deqe @? ne DE@Xo o :XăGdeqe#* o 88%}HX +A = pmatrix{col{ 0 # -1 } lcol{ 1 # 0 }} LBEquation Version 60Gdeqe#* !88&}HXA^50 LWEquation Version 60G ne )HB` C @onta deqe D E @o o :ăG onta )G deqe#* o 88'}HX(pmatrix{col{ -1 # 0 } lcol{ 0 # -1 }} LBEquation Version 60B`DE@RXăB`DE@_RXăB`DE@RXăB`DE@RXăB`DE@RXăB`DE@_RXăB`DE@RXăB` C  osg DE@ XăG osgCjZ %bGL` noc$noc$Z Z - ??????cer$cer$cer$&bL cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C2 -4 D E@OY Y B`U C P , deqe X deqe t 1` L,  deqe X i deqe X D lX|. ne DE7o o :Xă !!vXăGdeqe#* 8(o 88(}HX@UA = pmatrix{col{ 1 # x }lcol{ -2 # y }}, ~B = pmatrix{col{ 1 # -3 } lcol{ 2 # 3 }} LBEquation Version 60Gdeqe#* !88)}HX(AB)^2 = A^2 B^2 LWEquation Version 60Gdeqe#* e88*}HX`x, y LUEquation Version 60Gdeqe#* e88+}HX`x+y LUEquation Version 60G ne )HB`  C `onta 2 D E @ăG onta )B`DE@wRXăB`DE@RXăB`DE@RXăB`DE@7$RXăB`DE@w*RXăB`DE@0RXăB`DE@6RXăB`DE@7=RXăB`DE@wCRXăB`DE@IRXăB`DE@ORXăB`DE@7VRXăB` C@@ǀtǔ.... D'E@RXcB`DE@@RXcB`DE@ RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@%RXcB`DE@+RXcB`DE@2RXcB`DE@@8RXcB`DE@>RXcB`DE@DRXcB`DE@KRXcB`DE@@QRXcB` C@@ǀtǔ.... D'E@WRXcB`D'E@]RXcB`D'E@dRXcB`D'E@@jRXcB`D'E@pRXcB`D'E@vRXcB`D'E@}RXcB`D'E@@RXcB`D'E@RXcB`D'E@RXcB`D'E@RXcB`D'E@@RXcB`D'E@RXcB`D'E@RXcB` C@@ǀtǔ.... D'E@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB` C@@ǀtǔ.... D'E@@RXcB`D'E@RXcB`D'E@ RXcB`D'E@RXcB`D'E@@RXcB`D'E@RXcB`D'E@%RXcB`D'E@,RXcB`D'E@@2RXcB`D'E@8RXcB`D'E@>RXcB`D'E@ERXcB`D'E@@KRXcB`D'E@QRXcB`C  osg osg DE XăRXcG osgCj }HLcer$cer$Xaq w?"?V@ @ 0~1!@ CE ?u <V??7A}H  B `C`PD1 - 2. ,X Ű DE@@@PO aqaqG osgCj}HL` noc$noc$ aqB ??@ @&r?(F%0.}??noc$cer$cer$}L noc$ @&r???\\?c#{???cip$lle$Lcip$ @&r???\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$ @&r???\\?&gc#{?N[Q??+?+???!@H B`C1-1 DE` ` ` ` ` P8! and elsewhere.L cer$ @&r????PE???!@H B`C` DERX@RXO@ @ L cer$ @&r???????!@H B`DE@RXOXXB`C dloc osg DE@'J'J?XăG`dloc1F5G osgCjn'J‹}HLcer$cer$ZP,n'J]7%??ٛ?? `~:?@??!AË}H nnB ` C(2) ,X H DE@RXlB `$C 1) deqe , deqe | L deqe DE@@o o :XlGdeqe#* $o 88ŋ}HXA=pmatrix{a&b #c&dLBEquation Version 60Gdeqe#* o 88Ƌ}HXB=pmatrix{x&y#z&wLBEquation Version 60Gdeqe#* k*o 88Nj}HX)AB=pmatrix{ax+bz &```ay+bw#cx+dz&```cy+dwLBEquation Version 60B `DE@RXlB `8 C 2)|Ǭ-t4 Ȭ t(Ȭ, deqe X deqe t 1\. DEGo RXlo o :Xl&N*!!vXlGdeqe#* o 88}HX`A=` {pmatrix{a&b#c&d}}LBEquation Version 60Gdeqe#* 6!88}HX 8A ^{2} - LEFT ( a+d RIGHT ) A+ LEFT ( ad-bc RIGHT ) E=OLWEquation Version 60O ZZP,P,B` C  osg DE@LgXăG osgCjȋ}HL` noc$noc$q w@ ??????noc$cer$cer$ɋ}L noc$7zq 8 ?K@?? a/@???\\?c#{???cip$lle$Lcip$7ȯ z$t= ?K@?? a/@???\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$I 8!8! \\?@?@?fu@???\\?&gc#{?N[Q??+?+???!@H R B`C2-1 DE@` ` P P8! and elsewhere.L cer$@ @ ??t@?????PE???!@H  B`C` DE@RX O@ @ L cer$XXD,l?@?x@?iҼ@???????!@H )dB`DE@RXaOXXB`!C  P t( Ȭ, deqe deqe , DE@`: : /XGdeqe#* e88ʋ}HX P , {} Q LUEquation Version 60Gdeqe#* y!: 88ˋ}HX ;P+ 2 Q = LEFT ( { 1} atop {2 } { -3} atop {5 } RIGHT ) LCEquation Version 60B`$B}C deqe | ̹q` L, deqe X 1X D"E@l: : /XGdeqe#* ": 88̋}HX :P - Q = LEFT ( { 1} atop { -1} { 3} atop {-4 } RIGHT ) LCEquation Version 60Gdeqe#* e88͋}HXP + Q LUEquation Version 60B` C}Ci@? ne DE@xRXG ne )HB` C @onta deqe D E @eeăG onta )G deqe#* e88΋}HX4 LUEquation Version 60B`DE@Z~RXăB`C00 D&E@RXăB`DE@ڊRXăB`DE@RXăB`DE@ZRXăB`DE@RXăB`DE@ڣRXăB`DE@RXăB`DE@ZRXăB`DE@RXăB`DE@ڼRXăB`DE@RXăB`DE@ZRXăB` C  osg DE@gXăG osgCjqϋ}HL` noc$noc$qq 8 ??????noc$cer$cer$Ћ}L noc$7zq 8 ?K@?????\\?c#{???cip$lle$Lcip$7ȯ z$t= ?K@?????\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$I 8!8! \\?@?@????\\?&gc#{?N[Q??+?+???!@H  B`C2-2 DE@` `  P8! and elsewhere.L cer$@ @ ??t@?????PE???!@H @ B`C` DE@RX O@ @ L cer$XXD,l?@?x@????????!@H XB`DE@RXPVOXXB`2C@ deqe | ̹qXՔ  deqe X D lX|. ne DE@o o :XăGdeqe#* &o 88ы}HX{{pmatrix{pile{x#1} lpile{```````2#-1}}} {pmatrix{pile{1#y} lpile{```````3#-1}}} = {pmatrix{pile{````1``````#2} lpile{7#4}}}LBEquation Version 60Gdeqe#* e88ҋ}HX`x, y LUEquation Version 60G ne )HB` C @onta deqe D E @eeăG onta )G deqe#* e88Ӌ}HX x=3, y=-1 LUEquation Version 60B`C00 D&E@RXăB`DE@RXăB`DE@XRXăB`DE@RXăB`DE@RXăB`DE@RXăB`DE@XRXăB`DE@RXăB`DE@!RXăB`DE@(RXăB`DE@X.RXăB`DE@4RXăB`DE@:RXăB`DE@ARXăB`DE@XGRXăB`DE@RXcB`DE@@RXcB`DE@ RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@%RXcB`DE@+RXcB`DE@2RXcB`DE@@8RXcB`DE@>RXcB`DE@DRXcB`DE@KRXcB`DE@@QRXcB` C@@ǀtǔ.... D'E@WRXcB`DE@]RXcB`DE@dRXcB`DE@@jRXcB`DE@pRXcB`DE@vRXcB`DE@}RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB` C@@ǀtǔ.... D'E@@RXcB`D#E@RXcB`D#E@RXcB`D#E@RXcB`D#E@@RXcB`D#E@RXcB`D#E@RXcB`D#E@RXcB`D#E@@RXcB`D#E@RXcB`D#E@ RXcB`D#E@RXcB`D#E@@RXcB`D#E@RXcB`D#E@%RXcB` C  osg DE@,XcG osgCj,ԋ}HLcer$cer$d,Wk??::(@+M????!AՋ}H  ,B `C  ز TIP ƃ̹@.... D$E@RXx*O ddB`DE@/5RXcB`DE@o;RXcB`DE@ARXcB`DE@GRXcB`DE@RXăB` C  osg DE@@ XăG osgCjZ ֋}HL` noc$noc$Z Z - ??????cer$cer$cer$׋}L cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C2 - 1 D E@OY Y B`C 8 , DE@RXB` Q}C` deqe DE@: : /XGdeqe#* Q: 88؋}HXA = LEFT ( { 1} atop {0 } { 2} atop {-1 } RIGHT ) , {} B = LEFT ( { -1} atop {1 } { -2} atop {1 } RIGHT ) , {} C = LEFT ( { -1} atop { 2} { -2} atop {1 } RIGHT ) LCEquation Version 60B`$R}C deqe X deqe 1` L, DE@j'eeXGdeqe#* e88ً}HX x , {} y LUEquation Version 60Gdeqe#* e88ڋ}HXx A + y B = 3 C LUEquation Version 60B`S}C deqe X D lX|. ne DE@'.eeXGdeqe#* e88ۋ}HXx + y LUEquation Version 60G ne )HB` C @onta deqe D E @eeăG onta )G deqe#* e88܋}HX9 LUEquation Version 60B`C DE@4RXăB`DE@$;RXăB`DE@dARXăB`DE@GRXăB`DE@MRXăB`DE@$TRXăB` C  osg DE@dZ XăG osgCjZ ݋}HL` noc$noc$Z Z - ??????cer$cer$cer$ދ}L cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C2 - 2 D E@OY Y B`G C , deqe X deqe | ̹qXՔ  deqe X D lX|. ne DEio o :Xă%ueeXăGdeqe#* lo 88ߋ}HX*A= {pmatrix{pile{1#3} lpile{````1#````2}}}LBEquation Version 60Gdeqe#* !88}HXA^3 = xA +y E LWEquation Version 60Gdeqe#* e88}HX`x, y LUEquation Version 60G ne)HB` C `onta deqe D E @eeăG onta)G deqe#* e88}HX x=10, y=3 LUEquation Version 60B`C@ DE@@|RXăB`DE@RXăB`DE@RXăB`DE@RXăB`DE@@RXăB`DE@RXăB`DE@RXăB`DE@RXăB` C  osg DE@@ XăG osgCjZ }HL` noc$noc$Z Z - ??????cer$cer$cer$}L cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C2 - 3 D E@OY Y B`'C , deqe | L, deqe @? ne DE@o o :XăGdeqe#* o 88}HX +A = pmatrix{col{ 0 # -1 } lcol{ 1 # 0 }} LBEquation Version 60Gdeqe#* !88}HXA^50 LWEquation Version 60G ne)HB` C @onta deqe D E @o o :ăG onta)G deqe#* o 88}HX(pmatrix{col{ -1 # 0 } lcol{ 0 # -1 }} LBEquation Version 60B`DE@_RXăB`DE@RXăB`DE@RXăB`DE@RXăB`DE@_RXăB`DE@RXăB`DE@RXăB` C  osg DE@ XăG osgCjZ }HL` noc$noc$Z Z - ??????cer$cer$cer$}L cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C2 -4 D E@OY Y B`U C P , deqe X deqe t 1` L,  deqe X i deqe X D lX|. ne DEwo o :Xă>!!vXăGdeqe#* *o 88}HXZA= {pmatrix{pile{1#x} lpile{-2#```````y}}} ,`B= {pmatrix{pile{1#-3} lpile{`````2#`````3}}}LBEquation Version 60Gdeqe#* !88}HX(AB)^2 = A^2 B^2 LWEquation Version 60Gdeqe#* e88}HX`x, y LUEquation Version 60Gdeqe#* e88}HX`x+y LUEquation Version 60G ne)HB`  C `onta 2 D E @ăG onta)B`DE@RXăB`DE@RXăB`DE@7$RXăB`DE@w*RXăB`DE@0RXăB`DE@6RXăB`DE@7=RXăB`DE@wCRXăB`DE@IRXăB`DE@ORXăB`DE@7VRXăB`DE@RXcB` C@@ǀtǔ.... D'E@@RXcB`DE@ RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@%RXcB`DE@+RXcB`DE@2RXcB`DE@@8RXcB`DE@>RXcB`DE@DRXcB`DE@KRXcB`DE@@QRXcB`DE@WRXcB` C@@ǀtǔ.... D'E@]RXcB`D'E@dRXcB`D'E@@jRXcB`D'E@pRXcB`D'E@vRXcB`D'E@}RXcB`D'E@@RXcB`D'E@RXcB`D'E@RXcB`D'E@RXcB`D'E@@RXcB`D'E@RXcB`D'E@RXcB`D'E@RXcB` C@@ǀtǔ.... D'E@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB` C@@ǀtǔ.... D'E@RXcB`D'E@ RXcB`D'E@RXcB`D'E@@RXcB`D'E@RXcB`D'E@%RXcB`D'E@,RXcB`D'E@@2RXcB`D'E@8RXcB`C  osg osg DE XăRXcG osgCj ,}HLcer$cer$Xaq w?"?V@ @ 0~1!@ CE ?u <V??7A-}H  B `C@PD1 - 3. ʼn, Ž|( ) DE@@@PO aqaqG osgCj.}HL` noc$noc$ aqB ??@ @&r?(F%0.}??noc$cer$cer$/}L noc$ @&r???\\?c#{???cip$lle$Lcip$ @&r???\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$ @&r???\\?&gc#{?N[Q??+?+???!@H B`C1-1 DE` ` ` ` ` P8! and elsewhere.L cer$ @&r????PE???!@H B`C` DERX@RXO@ @ L cer$ @&r???????!@H B`DE@RXOXXB`C dloc osg DE@'J'J?XăG`dloc1F5G osgCjn'J0}HLcer$cer$ZP,n&J]7%??ٛ?? `~:?@??!A1}H nnB `C(1) ʼn, DE@RXlB `C 1) deqe DE@@  XlGdeqe#* tA 88\}HXYA= {pmatrix{a&b#c&d}} ```RARROW```A^-1 = { 1} over {ad-bc } pmatrix { d&-b #-c &``````a }LAEquation Version 60B `C  ʼn, tȬ pt : deqe DE@eeXlGdeqe#* e 88+HX D=ad-bc!=0LUEquation Version 60B `C 2) deqe DE@i8XlGdeqe#* 88}HX@ =LEFT ( A ^{n} RIGHT ) ^{-1} = LEFT ( A ^{-1} RIGHT ) ^{n}LYEquation Version 60B `C DE@!RXlB `C 3) deqe deqe DE@'!!vXlGdeqe#* !88}HX`A`=`BPB ^{-1}LWEquation Version 60Gdeqe#* !88}HX!RARROWA ^{n`} =BP ^{n} B ^{-1}LWEquation Version 60O ZZP,P,B` C  osg DE@LgXăG osgCj6}HL` noc$noc$q w@ ??????noc$cer$cer$7}L noc$7zq 8 ?K@?? a/@???\\?c#{???cip$lle$Lcip$7ȯ z t= ?K@?? a/@???\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$I 8!8! \\?@?@?fu@???\\?&gc#{?N[Q??+?+???!@H R B`C3-1 DE@` ` P P8! and elsewhere.L cer$@ @ ??t@?????PE???!@H  B`C` DE@RX O@ @ L cer$XXD,l?@?x@?iҼ@???????!@H )dB`DE@RXaOXXB`0C, deqe X ʼn, deqe X 1X i@?  ne D.+E@`o o :Gdeqe#* Jo 88i}HXA= pmatrix { 3~& 5#2~ &4 }LBEquation Version 60Gdeqe#* !88j}HX A^-1LWEquation Version 60G ne)HB` C `onta 0 D E @RXăG onta)B` C  deqe deqe D E @  Xă   XăG deqe#* ,6 88k}HX 9A^-1 = 1over34-52 pmatrix {`````` 4~& -5#-2~ & ``````3}LAEquation Version 60G deqe#*  88l}HX@ -= 1over2 pmatrix { ``````4~& -5# -2~&~````3 }LAEquation Version 60B` C  deqe D E @ XăG deqe#* r88m}HX 1= pmatrix {`````` 2~&- 5over2 # -1&``````3over2 }L:Equation Version 60B`C 0|, 1X i@ D E @=6RXăB` C  deqe D E @}<  XăG deqe#* [- 88n}HX*2+ LEFT ( - 5over2 RIGHT )+(-1)+ 3over2 =0LAEquation Version 60B`DE@lRXăB`C00 D&E@9sRXăB`DE@yyRXăB`DE@RXăB`DE@RXăB`DE@9RXăB`DE@yRXăB`DE@RXăB`DE@RXăB`DE@9RXăB`DE@yRXăB`DE@RXăB`DE@RXăB` C  osg DE@9gXăG osgCjq=}HL` noc$noc$qq 8 ??????noc$cer$cer$>}L noc$7zq 8 ?K@?????\\?c#{???cip$lle$Lcip$7ȯ z t= ?K@?????\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$I 8!8! \\?@?@????\\?&gc#{?N[Q??+?+???!@H  B`C3-2 DE@` `  P8! and elsewhere.L cer$@ @ ??t@?????PE???!@H @ B`C` DE@RX O@ @ L cer$XXD,l?@?x@????????!@H XB`DE@RXPVOXXB`KC` , deqe  deqe X ʼn,D 0 \  deqe X X ǔ?  ne DAEo o :Xă9wRXăGdeqe#* o 88v}HX pmatrix { a+1~&3 #b-1~ &2a }LBEquation Version 60Gdeqe#* e88w}HXaLUEquation Version 60Gdeqe#* e88x}HXbLUEquation Version 60G ne)HB`C @onta deqe D E @  XăG onta)G deqe#* `  88}HXb< {5} over {6}LAEquation Version 60B`C00 D)*)E@LLăB`DE@RXăB`DE@RXăB`DE@RXăB`DE@WRXăB`DE@RXăB`DE@ RXăB`DE@RXăB`DE@WRXăB`DE@RXăB`DE@#RXăB`DE@*RXăB`DE@W0RXăB`DE@6RXăB`DE@<RXăB`DE@CRXăB`DE@WIRXăB`DE@RXcB`DE@@RXcB`DE@ RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@%RXcB`DE@+RXcB`DE@2RXcB`DE@@8RXcB`DE@>RXcB`DE@DRXcB` C@@ǀtǔ.... D'E@KRXcB`DE@@QRXcB`DE@WRXcB`DE@]RXcB`DE@dRXcB`DE@@jRXcB`DE@pRXcB`DE@vRXcB`DE@}RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB` C@@ǀtǔ.... D'E@RXcB`D#E@RXcB`D#E@@RXcB`D#E@RXcB`D#E@RXcB`D#E@RXcB`D#E@@RXcB`D#E@RXcB`D#E@RXcB`D#E@RXcB`D#E@@RXcB`D#E@RXcB`D#E@ RXcB`D#E@RXcB`D#E@@RXcB` C  osg DE@XcG osgCj,B}HLcer$cer$d,Wk??::(@+M????!AC}H  ,B `C  ز TIP ƃ̹@.... D$E@RXx*O ddB`DE@(RXcB`DE@.RXcB`DE@/5RXcB`DE@o;RXcB` C  osg DE@A XcG osgCjZ D}HL` noc$noc$Z Z - ??????cer$cer$cer$E}L cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C3 - 1 D E@OY Y B`\C , deqe ʼn,D  Jij] XՔ  deqe t() deqe D ̹q\. t L, deqe X @?  ne DEXo o :X/!!vXGdeqe#* 'o 88}HX pmatrix { 8x-2~&3 #x+2~ &x }LBEquation Version 60Gdeqe#* Xe88}HXxLUEquation Version 60Gdeqe#* !88}HX  8x^2 +bx+c=0LWEquation Version 60Gdeqe#* e88}HXb+cLUEquation Version 60G ne)HB` C onta -11 D E @RXăG onta)B`D E @@RXăB`C DE@"RXăB`DE@(RXăB`DE@/RXăB`DE@X5RXăB`DE@;RXăB`DE@ARXăB` C  osg DE@H XăG osgCjZ K}HL` noc$noc$Z Z - ??????cer$cer$cer$L}L cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C3 - 2 D E@OY Y B`SC` t( Ȭ, deqe deqe | ̹q` L, deqe D deqe @ deqe \ д|. ne D)EpW!!văE%_RXăGdeqe#* 9e88}HXA LUEquation Version 60Gdeqe#* o!88}HX@ A+A^-1 = E LWEquation Version 60Gdeqe#* U!88}HX@A^2 LWEquation Version 60Gdeqe#* 9e88}HX A LUEquation Version 60Gdeqe#* e88 }HX E LUEquation Version 60G ne)HB` C @onta deqe D E @!!văG onta)G deqe#* !88!}HX A^2 = A-E LWEquation Version 60B`DE@eeRXăB`DE@kRXăB`DE@qRXăB`DE@%xRXăB`DE@e~RXăB` C  osg DE@ XăG osgCjZ Q}HL` noc$noc$Z Z - ??????cer$cer$cer$R}L cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C3- 3 D E@OY Y B`RC@ XX  deqe X , deqe ʼn,D ij] XՔ  deqe X X ǔ? ne DEo o :Xă2ğeeXăGdeqe#* e88}HXkLUEquation Version 60Gdeqe#* $o 88}HXpmatrix{k-a&~~k+2#-2&~~k}LBEquation Version 60Gdeqe#* e88}HXaLUEquation Version 60G ne)HB`C onta  `$ D E @RăG onta)B` C `ʼn,t tȬ` Ɣ deqe tǴ| X\ deqe deqe 0|, deqe \ t() deqe t D  JD| X\ ļ deqe tǴ| \. deqe , deqe , deqe deqe D E eeă5 !eă&!!vă/!!vă\ !eău'!!vă&.eeăG deqe#* fe88}HX ad-bc !=0LUEquation Version 60G deqe#* .e88}HXk (k-a)- (k+2 )(-2)!= 0LUEquation Version 60G deqe#* "!88}HXk^2 -ak +2k +4 != 0LWEquation Version 60G deqe#* e88 }HXkLUEquation Version 60G deqe#* #!88 }HXk^2 - (a-2)k+4 =0LWEquation Version 60G deqe#* e88 }HX D< 0LUEquation Version 60G deqe#* a!88 }HX(a-2)^2 -16 < 0LWEquation Version 60G deqe#* 3!88 }HX`a^2 -4a -12< 0LWEquation Version 60G deqe#* e88}HX@ (a+2)(a+6) <0LUEquation Version 60G deqe#* Re88}HXtherefore-2 < a<6LUEquation Version 60B`C@ DE@RXăB`DE@RXăB`DE@RXăB`DE@ARXăB`DE@RXăB`DE@RXăB` C  osg DE@ XăG osgCjZ V}HL` noc$noc$Z Z - ??????cer$cer$cer$W}L cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C3 -4 D E@OY Y B`< C P , deqe X deqe | ` L, deqe @? ne DEYo o :Xă& !!vXăGdeqe#* >&o 88}HX QA= {pmatrix{pile{2#0} lpile{````1#````3}}} ,`P= {pmatrix{pile{1#0} lpile{-1#-1}}}LBEquation Version 60Gdeqe#* !88}HX  A= PBP^-1 LWEquation Version 60Gdeqe#* !88}HX`B^3 LWEquation Version 60G ne)HB` C onta deqe D E @o o :ăG onta)G deqe#* , o 88}HX@%{pmatrix{pile{8#2``} lpile{``0#``7}}}LBEquation Version 60B`DE@RXăB`DE@RXăB`DE@RXăB`DE@YRXăB`DE@RXăB`DE@ RXăB`DE@RXăB`DE@YRXăB`DE@ RXăB` C@@ǀtǔ.... D'E@&RXăB`DE@-RXăB`DE@Y3RXăB`DE@9RXăB`DE@?RXăB`DE@FRXăB`DE@YLRXăB`DE@RRXăB`DE@RXcB`DE@@RXcB`DE@ RXcB`DE@RXcB`DE@RXcB`DE@@RXcB` C@@ǀtǔ.... D'E@%RXcB`D'E@+RXcB`D'E@2RXcB`D'E@@8RXcB`D'E@>RXcB`D'E@DRXcB`D'E@KRXcB`D'E@@QRXcB`D'E@WRXcB`D'E@]RXcB`D'E@dRXcB`D'E@@jRXcB`D'E@pRXcB`D'E@vRXcB` C@@ǀtǔ.... D'E@}RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB` C@@ǀtǔ.... D'E@@RXcB`D'E@RXcB`D'E@RXcB`D'E@RXcB`D'E@@RXcB`D'E@RXcB`D'E@RXcB`D'E@RXcB`D%E@@cB`D%E@ cB`D%E@cB`D%E@`cB`D%E@%cB`D%E@ /cB`D%E@8cB`D%E@AcB`D%E@@KcB`D%E@TcB`C  osg osg DE XăRXcG osgCj }HLcer$cer$Xaq w?"?V@ @ 0~1!@ CE ?u <V??7A}H  B `C@PD1 - 3. ʼn, Ž|( ) DE@@@PO aqaqG osgCj}HL` noc$noc$ aqB ??@ @&r?(F%0.}??noc$cer$cer$}L noc$ @&r???\\?c#{???cip$lle$Lcip$ @&r???\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$ @&r???\\?&gc#{?N[Q??+?+???!@H B`C1-1 DE` ` ` ` ` P8! and elsewhere.L cer$ @&r????PE???!@H B`C` DERX@RXO@ @ L cer$ @&r???????!@H B`DE@RXOXXB`C dloc osg DE@'J'J?XăG`dloc1F5G osgCjn'J}HLcer$cer$ZP,n&J]7%??ٛ?? `~:?@??!A}H nnB ` C(2) ʼn, Ž|() 1) deqe DEo RXl@o o :XlGdeqe#* 3o 88}HXW{cases{ax+by=0&#cx+dy=0&}} ` RARROW {pmatrix{a&b#c&d}} {pmatrix{x#y}} = {pmatrix{0#0}}LBEquation Version 60B `$C `$ deqe deqe \ X t| . DE@eeXlGdeqe#* e88}HXad-bc`!=0 ``RARROW LUEquation Version 60Gdeqe#* e88}HX@x=y=0LUEquation Version 60B `'C a$ deqe deqe txX t| 4ˆ . DE@eeXlGdeqe#* Ke88}HXad-bc=0``RARROWLUEquation Version 60Gdeqe#* e88}HX@x=y=0LUEquation Version 60B `DE@RXlB `C 2) deqe DE@%o o :XlGdeqe#* 3o 88}HXW{cases{ax+by=p&#cx+dy=q&}} ` RARROW {pmatrix{a&b#c&d}} {pmatrix{x#y}} = {pmatrix{p#q}}LBEquation Version 60B `C@ `$ deqe \ X t| . DE@1eeXlGdeqe#* e88}HXad-bc`!=0 ``RARROW LUEquation Version 60B `C a$ deqe   DE@E8eeXlGdeqe#* Ke88}HXad-bc=0``RARROWLUEquation Version 60O ZZP,P,B` C  osg DE@LgXăG osgCj}HL` noc$noc$q w@ ??????noc$cer$cer$}L noc$7zq 8 ?K@?? a/@???\\?c#{???cip$lle$Lcip$7ȯ z t= ?K@?? a/@???\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$I 8!8! \\?@?@?fu@???\\?&gc#{?N[Q??+?+???!@H R B`C3-3 DE@` ` P P8! and elsewhere.L cer$@ @ ??t@?????PE???!@H  B`C` DE@RX O@ @ L cer$XXD,l?@?x@?iҼ@???????!@H )dB`DE@RXaOXXB`OC Ž) deqe | ,D tǩX дt deqe t. t L, t(Ȭ, deqe | lX|. ne DE`o o :Xă'leeXăGdeqe#* o 88}HXcases{4x-y=-1&#x-3y=9&}LBEquation Version 60Gdeqe#* mo 88}HX` 6X LEFT ( pile{x#y} RIGHT ) = LEFT ( pile{-1#9} RIGHT )LBEquation Version 60Gdeqe#* 9e88}HXXLUEquation Version 60G ne)HB`C onta  deqe D E @o o :XăG onta)G deqe#* o 88}HX`X= pmatrix{4~&-1#1~&-3}LBEquation Version 60B` C @ deqe deqe D E  o o :Xă o o :XăG deqe#* Co 88}HXscases{4x-y=-1&#x-3y=9&} `````~ LRARROW ~~ pmatrix{4&~-1#1&~-3} LEFT ( pile{x#y} RIGHT ) = LEFT ( pile{-1#9} RIGHT )LBEquation Version 60G deqe#* o 88}HX"THEREFORE `X= pmatrix{4~&-1#1~&-3}LBEquation Version 60B`DE@zsRXăB`C00 D&E@yRXăB`DE@RXăB`DE@:RXăB`DE@zRXăB`DE@RXăB`DE@RXăB`DE@:RXăB`DE@zRXăB`DE@RXăB`DE@RXăB`DE@:RXăB`DE@zRXăB` C  osg DE@gXăG osgCjq}HL` noc$noc$qq 8 ??????noc$cer$cer$}L noc$7zq 8 ?K@?????\\?c#{???cip$lle$Lcip$7ȯ z t= ?K@?????\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$I 8!8! \\?@?@????\\?&gc#{?N[Q??+?+???!@H  B`C3-4 DE@` `  P8! and elsewhere.L cer$@ @ ??t@?????PE???!@H @ B`C` DE@RX O@ @ L cer$XXD,l?@?x@????????!@H XB`DE@RXPVOXXB`iC deqe \ Ž) deqe t t|  Jp 4ˆ ι@ t| ij] XՔ  deqe X deqe X \͌D lX|. ne DE1o o :Xă&eeXă[RXăGdeqe#* e88}HX`x, y LUEquation Version 60Gdeqe#* 6o 88}HX 4cases{ (a-2)x+ by=a-1 # (a-1)x+ (a+b-1)y = a-2 }} LBEquation Version 60Gdeqe#* He88}HXa, b LUEquation Version 60Gdeqe#* e88}HX b LUEquation Version 60G ne)HB` C @onta deqe D E @  ăG onta)G deqe#*  88}HX  - 1 over 4 LAEquation Version 60B`DE@RXăB`C00 D)*)E@5LLăB`DE@RXăB`DE@URXăB`DE@ RXăB`DE@RXăB`DE@RXăB`DE@URXăB`DE@$RXăB`DE@*RXăB`DE@1RXăB`DE@U7RXăB`DE@=RXăB`DE@CRXăB`DE@JRXăB`DE@RXcB`DE@@RXcB`DE@ RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@%RXcB`DE@+RXcB`DE@2RXcB`DE@@8RXcB`DE@>RXcB`DE@DRXcB`DE@KRXcB`DE@@QRXcB`DE@WRXcB` C@@ǀtǔ.... D'E@]RXcB`DE@dRXcB`DE@@jRXcB`DE@pRXcB`DE@vRXcB`DE@}RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB` C@@ǀtǔ.... D'E@RXcB`D#E@RXcB`D#E@RXcB`D#E@@RXcB`D#E@RXcB`D#E@RXcB`D#E@RXcB`D#E@@RXcB`D#E@RXcB`D#E@ RXcB`D#E@RXcB`D#E@@RXcB`D#E@RXcB`D#E@%RXcB`D#E@,RXcB` C  osg DE@@2XcG osgCj,}HLcer$cer$d,Wk??::(@+M????!A}H  ,B `C  ز TIP ƃ̹@.... D$E@RXx*O ddB`DE@o;RXcB` C  osg DE@ XăG osgCjZ }HL` noc$noc$Z Z - ??????cer$cer$cer$}L cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C3 - 5 D E@OY Y B`NC Ž) deqe t \ X t| ptt deqe  deqe | L, deqe X @? ne DEXo o :Xă)eeXăGdeqe#* %o 88}HX@cases{ax+2y=5&#3x+(1+a)y=-3&}LBEquation Version 60Gdeqe#* e88}HX`a != mLUEquation Version 60Gdeqe#* e88}HX`a != nLUEquation Version 60Gdeqe#* I e 88}HXm+nLUEquation Version 60G ne)HB`  C @onta-1 deqe deqe 11D ,\ дt deqe tǃt $ \ X t| $t ʼn,t tȬt| \. deqe deqe deqe  deqe deqe D  E o RXă @o o :Xăo RXă(o o :Xc1 RXcO!RXcSG!eXc\!!vXce}&eeXcy:-eeXcG onta)G deqe#* %o 88+HX@cases{ax+2y=5&#3x+(1+a)y=-3&}LBEquation Version 60G deqe#* 5 e88+HX`CDOTS CDOTS 11LUEquation Version 60G deqe#* !o 88+HXWLEFT ( matrix{a&~2#3&~1+a} RIGHT ) LEFT ( pile{x#y} RIGHT ) = LEFT ( pile{5#-3} RIGHT )LBEquation Version 60G deqe#* e88+HX@ a(1+a)-6 != 0LUEquation Version 60G deqe#* :!88+HX"a ^{2} +a-6 != 0,~(a-2)(a+3) != 0LWEquation Version 60G deqe#* e 88+HXTHEREFORE a != 2`LUEquation Version 60G deqe#* e!88+HX`a!= -3LUEquation Version 60G deqe#* ,e"88+HXTHEREFORE m+n=2+(-3)=-1LUEquation Version 60B`C DE@!RXăB`DE@(RXăB`DE@\.RXăB`DE@4RXăB`DE@:RXăB`DE@ARXăB` C  osg DE@\G XăG osgCjZ }HL` noc$noc$Z Z - ??????cer$cer$cer$}L cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C3 - 6 D E@OY Y B`iC , deqe X ʼn,@ deqe t.  deqe \ Ž) deqe   deqe ` L, deqe X \@? ne DEVo o :Xă/{b+ + XăGdeqe#* o 88+HX A= pmatrix { a~&~b # c~&~d }LBEquation Version 60Gdeqe#* Ao 88+HX $A^{ -1 }= pmatrix { 1~&~-2 # 3~&~~1}LBEquation Version 60Gdeqe#* e 88+HX x,~yLUEquation Version 60Gdeqe#* + 88+HX@cases{ ax+by=t # &cx+dy=t^2 }LAEquation Version 60Gdeqe#* we88+HXtLUEquation Version 60Gdeqe#* e88+HXx+yLUEquation Version 60G ne)HB` C `onta 4 D E @RXcG onta)B`C  deqe  deqe t\ D E @@+ + XcG deqe#* + 88+HX cases{ ax+by=t # cx+dy=t^2 }LAEquation Version 60G deqe#* + 88+HX Ipmatrix { a~~&~b # c~~&~d } pmatrix { ~x` # ~y` }=pmatrix { ~t` # ~t^2` }LAEquation Version 60B`C  deqe D E @ XcG deqe#* = 88+HXpmatrix { ~x`# ~y` }= {pmatrix { a~~&~b # c~~&~d }}^-1 pmatrix { ~t` # ~t^2` }=pmatrix { 1~~&~-2 # 3~~&~~1 } pmatrix { ~t` # ~t^2` LBEquation Version 60B`C  deqe D E @ XcG deqe#* v 88+HX@=pmatrix { ~t-2t^2 # 3t+t^2 }L@Equation Version 60B`C  deqe ( deqe ) D E @,!!vXcG deqe#* .!88+HX"THEREFORE~ x=t-2t^2 ,~~y=3t+t^2 LWEquation Version 60G deqe#* we88+HXtLUEquation Version 60B`C  deqe D E @\4!!vXcG deqe#* ?!88+HX`&THEREFORE~~ x+y=-t^2 + 4t =-(t-2)^2 +4LWEquation Version 60B`4C  0|, deqe deqe | L, \ deqe | . D E @;eeXcG deqe#* e88+HXx+yLUEquation Version 60G deqe#* e88+HXt=2LUEquation Version 60G deqe#* e88+HX4LUEquation Version 60B`DE@nRXăB`DE@>uRXăB`DE@~{RXăB`DE@RXăB` C  osg DE@ XăG osgCjZ }HL` noc$noc$Z Z - ??????cer$cer$cer$}L cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C3- 7 D E@OY Y B`U C deqe \ Ž) deqe t t|  JD L,  deqe X deqe X \͌@?  ne DEVo eXăo o :XăکeeXăGdeqe#* e#88+HX x,~yLUEquation Version 60Gdeqe#* "9o $88+HX*cases{(a-2)x+by=a-4&#(a-1)x+(a+b-1)y=a-3&}LBEquation Version 60Gdeqe#* ne%88+HX a,~bLUEquation Version 60Gdeqe#* e&88,HXbLUEquation Version 60G ne)HB`y C onta deqe deqe deqe ȴ Ž)t t|  J<\ deqe deqe deqe 0|, lXՔ deqe X \͌@ deqe t. D E   Xcl o o :Xc3o o :Xc##RXc::*!eXcC0 !vXcLp8 ZXcUE  XcG onta)G deqe#*  .88,HX`- {1} over {4}LAEquation Version 60G deqe#* "9o '88,HX*cases{(a-2)x+by=a-4&#(a-1)x+(a+b-1)y=a-3&}LBEquation Version 60G deqe#* 5o (88,HXkLRARROW `~ LEFT ( matrix{a-2&~b#a-1&~a+b-1} RIGHT ) LEFT ( pile{x#y} RIGHT ) = LEFT ( pile{a-4#a-3} RIGHT )LBEquation Version 60G deqe#* 7e)88,HX@(a-2)(a+b-1)-b(a-1)=0LUEquation Version 60G deqe#* !*88,HXa ^{2} -3a-b+2=0LWEquation Version 60G deqe#* : +88,HX` NTHEREFORE b=a ^{2} -3a+2= LEFT ( a- {3} over {2} RIGHT ) ^{2} - {1} over {4}LDEquation Version 60G deqe#* e,88,HXbLUEquation Version 60G deqe#*  -88,HX`- {1} over {4}LAEquation Version 60B`DE@RXăB`DE@׶RXăB`DE@RXăB`DE@WRXăB`DE@RXăB` C  osg DE@ XăG osgCjZ }HL` noc$noc$Z Z - ??????cer$cer$cer$}L cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C3 -8 D E@OY Y B`TC t() deqe X P D deqe | ` L, Ž) deqe deqe txX t| pt@? ne DE/!!vXă*o o :XăGdeqe#* q!/88,HX@ x^2 + 2x -2=0LWEquation Version 60Gdeqe#* e088,HX@a,~~dLUEquation Version 60Gdeqe#* co 188,HX Gpmatrix { a~&~b # c~&~d } pmatrix { ~x` # ~y` }= pmatrix { ~x` # ~y` }LBEquation Version 60Gdeqe#* e288,HX@x=y=0LUEquation Version 60G ne)H B`C @onta deqe D E @eeXcG onta)G deqe#* e=88!,HX bc=1LUEquation Version 60B`#C ` t() deqe X P t deqe t\ D E @!!vXcG deqe#* q!388,HX x^2 +2x-2=0LWEquation Version 60G deqe#* e488,HX a,~dLUEquation Version 60B` C  deqe l  & `2 D E @6eeXcG deqe#* @$e588,HX@ a+d=-2,~ad=-2LUEquation Version 60B`C  deqe  D E @o o :XcG deqe#* o 688,HX Gpmatrix { a~~&~b # c~~&~d } pmatrix { ~x` # ~y` }=pmatrix { ~x` # ~y` }LBEquation Version 60B`C  deqe D E @ o o :XcG deqe#* 2*o 788,HX Kpmatrix { a-1~~&~b # c~~&~d-1 }pmatrix { ~x` # ~y` }= pmatrix { ~0` # ~0` }LBEquation Version 60B` C  t Ž)t deqe txX t| \ D E @,eeXcG deqe#* e888,HX@x=y=0LUEquation Version 60B`C  deqe D E @>3eeXcG deqe#* &e988,HX(a-1)(d-1)-bc=0LUEquation Version 60B`( C  deqe    & a2 D E 9eeXc&@RXcG deqe#* k.e:88,HXTHEREFORE~ ad-(a+d)+1-bc=0LUEquation Version 60B`C  t L, `2D a2 Xt D E @FRXcB`& C  deqe   deqe D E @8MeeXcG deqe#* !e;88,HX -2+2+1-bc=0LUEquation Version 60G deqe#* e<88 ,HX`THEREFORE~bc=1LUEquation Version 60B`DE@oRXăB`DE@RXăB`DE@RXăB`DE@/RXăB`DE@o RXăB`DE@RXăB`DE@RXăB`DE@/RXăB`DE@o$RXăB` C@@ǀtǔ.... D'E@*RXăB`DE@0RXăB`DE@/7RXăB`DE@o=RXăB`DE@CRXăB`DE@IRXăB`DE@/PRXăB`DE@oVRXăB`DE@RXcB`DE@@RXcB`DE@ RXcB`DE@RXcB`DE@RXcB`DE@@RXcB` C@@ǀtǔ.... D'E@%RXcB`D'E@+RXcB`D'E@2RXcB`D'E@@8RXcB`D'E@>RXcB`D'E@DRXcB`D'E@KRXcB`D'E@@QRXcB`D'E@WRXcB`D'E@]RXcB`D'E@dRXcB`D'E@@jRXcB`D'E@pRXcB`D'E@vRXcB` C@@ǀtǔ.... D'E@}RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB` C@@ǀtǔ.... D'E@@RXcB`D'E@RXcB`D'E@RXcB`D'E@RXcB`D'E@@RXcB`D'E@RXcB`D'E@RXcB`D'E@RXcB`D%E@@cB`D%E@ cB`D%E@cB`D%E@`cB`D%E@%cB`D%E@ /cB`D%E@8cB`D%E@AcB`D%E@@KcB`D%E@TcB`C  osg osg DE XăRXcG osgCj G}HLcer$cer$Xaq w?"?V@ @ 0~1!@ CE ?u <V??7AH}H  B `CPD1 - 4. ,  DE@@@PO aqaqG osgCjI}HL` noc$noc$ aqB ??@ @&r?(F%0.}??noc$cer$cer$J}L noc$ @&r???\\?c#{???cip$lle$Lcip$ @&r???\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$ @&r???\\?&gc#{?N[Q??+?+???!@H B`C1-1 DE` ` ` ` ` P8! and elsewhere.L cer$ @&r????PE???!@H B`C` DERX@RXO@ @ L cer$ @&r???????!@H B`DE@RXOXXB`C dloc osg DE@^^OXăG`dloc1F5G osgCjqTNK}HLcer$cer$ZP,^w/??HX!@f@@??!AL}H nB ` C  lbt DE@:R:REXtG lbt#*@PAa:HMH u uB`C1. DE@RXqB`C 1) : (-) ()<\ и ij DE@@RXqB` C` 2)\@ \ DE@ RXqB` C  \ : DE@RXqB`C ½\ : t  JŔ \ DE@RXqB`C \ : ‘ ]t @ \ DE@@RXqB`C Œ\ : t  JŔ \ DE@%RXqB` C  3)\Փ0 DE@+RXqB`VC \Փ0 Xt ‘ ]D x\ -@ 䴴$Ɣ X t Ǵ| X\  X t Ű Ǵ| \. DE2RXq%@8RXqH>RXqB`=C , ‘ ]t x \Փ0 Xt ‘ ] Ű X  P @t. DEDRXq&KRXqHu uB` C@2.@ , DE@RXqB`*C@ 1)xȉ, : P - t| ŰXՔ X | 1<\ XՔ , DE@RXq' RXqB`C@ X -t deqe (, deqe )t P - deqe @ deqe | ŰXՔ X  deqe | L, deqe ( Ȭ, deqe | X xȉ,t| \. DEFFXq5^FFXqq!RXqGdeqe#* FB88b:HXV _{i}LGEquation Version 60Gdeqe#* eC88c:HXi=1,`2,`3,`cdots,nLUEquation Version 60Gdeqe#* FD88d:HXV_iLGEquation Version 60Gdeqe#* eFE88e:HXV_jLGEquation Version 60Gdeqe#* FF88f:HX a_ijLGEquation Version 60Gdeqe#* XeG88g:HXnLUEquation Version 60Gdeqe#* HFH88h:HX`A= LEFT ( a_ij RIGHT )LGEquation Version 60B`C@ 2)xȉ,X \X  DE@<(RXqB`NC deqe X deqe 1@ - deqe  deqe \ 8t deqe x \X t. DE|.FFvXqG6RXqGdeqe#* U!I88i:HXA^kLWEquation Version 60Gdeqe#* eJ88j:HX@LEFT ( i, ``j RIGHT )LUEquation Version 60Gdeqe#* FK88k:HXV_iLGEquation Version 60Gdeqe#* eFL88l:HXV_jLGEquation Version 60Gdeqe#* eM88m:HXkLUEquation Version 60B `DE@TRXtO ZZP,P,B` C  osg DE@[`gXăG osgCjV}HL` noc$noc$q w@ ??????noc$cer$cer$W}L noc$7zq 8 ?K@?? a/@???\\?c#{???cip$lle$Lcip$7ȯ z t= ?K@?? a/@???\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$I 8!8! \\?@?@?fu@???\\?&gc#{?N[Q??+?+???!@H R B`C4-1 DE@` ` P P8! and elsewhere.L cer$@ @ ??t@?????PE???!@H  B`C` DE@RX O@ @ L cer$XXD,l?@?x@?iҼ@???????!@H )dB`DE@RXaOXXB`QC -X  6x   - Ű X  4, 3, 3, 3, 3, 2 | L, t X X ”? ne DEtRXă3{RXăG ne)HB`  C `onta 9 D E @cG onta)B`C00 D&E@RRXăB`C@ DE@RXăB`DE@ҍRXăB`DE@RXăB`DE@RRXăB`DE@RXăB`DE@ҦRXăB`DE@RXăB`DE@RRXăB`DE@RXăB`DE@ҿRXăB`DE@RXăB` C  osg DE@RgXăG osgCjq^}HL` noc$noc$qq 8 ??????noc$cer$cer$_}L noc$7zq 8 ?K@?????\\?c#{???cip$lle$Lcip$7ȯ z t= ?K@?????\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$I 8!8! \\?@?@????\\?&gc#{?N[Q??+?+???!@H  B`C4-2 DE@` `  P8! and elsewhere.L cer$@ @ ??t@?????PE???!@H @ B`C` DE@RX O@ @ L cer$XXD,l?@?x@????????!@H XB`DE@RXPVOXXB`cC` osg $x  - deqe  - deqe \ \X | lX|.(, \ ɘ -@ ɘ JŔ.) ne DE eeX I#eeX IBCRX IWRX IG osg"  I:5>]:HL@ cip$cip$aY:5 d??333333?333333???U aaYYaYGdeqe#* 9e?88_:HXALUEquation Version 60Gdeqe#* e@88`:HXELUEquation Version 60G ne)HB`  C onta 9 D E @cG onta)B`DE@RX IB`DE@RX IB`DE@CRX IB`DE@ RX IB`DE@RX IB`DE@RXăB`DE@C RXăB`DE@&RXăB`C00 D&E@,RXăB`DE@3RXăB`DE@C9RXăB`DE@?RXăB`DE@ERXăB`DE@RXcB`DE@@RXcB`DE@ RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@%RXcB`DE@+RXcB`DE@2RXcB`DE@@8RXcB`DE@>RXcB`DE@DRXcB`DE@KRXcB`DE@@QRXcB`DE@WRXcB`DE@]RXcB`DE@dRXcB` C@@ǀtǔ.... D'E@@jRXcB`DE@pRXcB`DE@vRXcB`DE@}RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB` C@@ǀtǔ.... D'E@RXcB`D#E@RXcB`D#E@@RXcB`D#E@RXcB`D#E@RXcB`D#E@RXcB`D#E@@RXcB`D#E@RXcB`D#E@ RXcB`D#E@RXcB`D#E@@RXcB`D#E@RXcB`D#E@%RXcB`D#E@,RXcB` C  osg DE@@2XcG osgCj,d}HLcer$cer$d,Wk??::(@+M????!Ae}H  ,B `C  ز TIP ƃ̹@.... D$E@RXx*O ddB`DE@o;RXcB`DE@ARXcB`DE@GRXcB` C  osg DE@ XăG osgCjZ f}HL` noc$noc$Z Z - ??????cer$cer$cer$g}L cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C4 - 1 D E@OY Y B`C DŘ X xȉ,D lX|. ne DE@XRXăG ne)HB`  C @onta D E @cG onta)B` C  osg DE@V1V1)XăG osg#*81V1On:HL@ cip$cip$pbb81V1 ??????U pbpbbbpbbB`DE@FIRXăB`DE@ORXăB` C  osg DE@U XăG osgCjZ l}HL` noc$noc$Z Z - ??????cer$cer$cer$m}L cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`  DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C4 - 2 D E@OY Y B`C xȉ,t L @ | $|. 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