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Determinant & Matrices TS Eng

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(14)

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(15)

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(16)

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(17)

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(18)

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(19)

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(20)

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HKI = HXI , HEI a H;I G YlLL ,GGbHT.1IZ

_5= 63 P -$ * = ! H &'()*+J I.5)5 W!P < 6 *2/ /#0 HPI < +D 0"#2 @1(7# 0"*0 /#0 HP L% G K$

YlLL ,GGbD , (80 (3 AGZ _5b K < .  ' + /  ' + G + * D X < " 9 3  . / G + + * D n < " 9 3  D r < G G *, D *2/ q -$ * .()8&2 7#.0(1 -5#5 q < V % ? 5 63Kq < n "*$ -23-2-0#)%&*2%$()80-(2D 0"#2@1(7# 0"*0 Xq< r .*22(0 "*7# * 82-J8# $()80-(25 63 3810"#1  *3/ GD 0"#2 @1(7# 0"*0 Xq < r "*$ 2( $()80-(25 YlLL ,GGbD b (80 (3 AGZ _5a K < b , G + + G G G + D 6 < + G G G + G G G + *2/KK ,< HK .K /6I A + , D 0"#2 0"# 7*)8# (3 . *2/ / *1#

(21)

 <,=,>?@A*A= B C*=>@D,E   F-5G  _5A 63 [ < , = , + , + , = D K < G+ ++ *2/ _ < [K[K[!*2/ ? < [!_,GGa[D 0"#2 ? -$ #J8*) 0(

HKI G+ ,GGa+ HXI b ,GGa,GGa = b AG+a,GGa =

HEI , += , + =

b +

H;I ,,GGa= ,,GGa=

b +

YlLL ,GGa HT.1##2-29IZ )=JaS?R?NO1=N 0/ PK?O>1=NO2 _5d + , = G + , G G + K D n+D n,*2/ n=*1# .()8&2$ &*01-.#$ $*0-$3%-295Kn+< G G + B KnKn,< G = , DKn=< + = ,

*2/n-$= ! H &'()*+ I./-5 6/1=&#- ')5 ]+D n,D n=0"#2 *2$4#1 0"# 3())(4-29 J8#$0-(2$ H*I !"# 7*)8# (3 p n p - $

HKI = HXI H   HEI =g, H;I ,

H'I !"# $8& (3 #)#&#20$ (3 n ,-$

HKI , HXI G HEI + H;I =

H.I !"# 7*)8# (3  G , = n G , = -$

HKI a HXI ag, HEI b H;I =g,

YlLL ,GGAD a &*1U$ #*."Z _5c P*0." 0"# $0*0#&#20$ g L?@1#$$-(2 -2 )=LKJN]& 4-0" 0"# $0*0#&#20$ g L?@1#$$-(2$ -2 )=LKJN]&& *2/

-2/-.*0# %(81 *2$4#1 '%/*1U#2-29 0"#*@@1(@1-*0# '8'')#$ -2 0"# b ! ^ &'()*+ ?*45# *# _W`$ )=LKJN]& )=LKJN]&& HKI !"# &-2-&8& 7*)8# (3  , ? b ? , ?, -$ H[I G

HXI W#0 K *2/ X ' # = ! H &'()*65- /2 )5'1 #=&B5)-J H_I +

4"#1#K -$ $%&&#01-.D X -$ $U#4k$%&&#01-.D *2/

HK > XIHK 9% G A: 9%A: a 9%$ L2 A:9%0< H ,%U KXD 4"#1# HKXI0 -$ 0"# 01*2$@($# (3 0"# &*01-?KXD 0"#2 0"# @($$-')# 7*)8#$ (3 U *1#

HEI W#0 * < )(9=)(9=,5K2 -20#9#1 U $*0-$3%-29 + m ,H U  = *I m ,D &8$0 '# HCI , )#$$ 0"*2 H;I 63 $-2 < .($ D 0"#2 0"# @($$-')# 7*)8#$ (3  , + *1# HTI = YlLL ,GGcD AZ

(22)

 <,=,>?@A*A= B C*=>@D,E   F--G  )=JaS?R?NO1=N 0/ PK?O>1=NO2

_5` W#0 '#0"# $#0 (3*))= ! H ->&&5()*6 &'()*65- '11 /2 I./-5 ')5 5*(.5) K /) ,$ R*45 /2 (.5-5 5#()*5-*1# + *2/ 3(81 (3 0"#& 5#()*5-*1# G5

H*I !"# 28&'#1 (3 &*01-.#$ -2 -$

HKI +, HXI A HEI ` H;I =

H'I !"# 28&'#1 (3&*01-.#$ K-2 3(1 4"-." 0"# $%$0#& (3 )-2#*1 #J8*0-(2$

G G + V % ? K "*$ * 82-J8# $()80-(2D -$ HKI )#$$ 0"*2 b HXI *0 )#*$0 b '80 )#$$ 0"*2 d HEI *0 )#*$0 d '80 )#$$ 0"*2 +G H;I *0 )#*$0 +G

H.I !"# 28&'#1 (3&*01-.#$ K-2 3(1 4"-." 0"# $%$0#& (3 )-2#*1 #J8*0-(2$

G G + V % ? K -$ -2.(2$-$0#20D -$

HKI G HXI &(1# 0"*2 , HEI , H;I + YlLL ,GG`D b>b>bZ

_5+GH*I!"# 28&'#1 (3 = ! H &'()*65- : I./-5 5#()*5- ')5 5*(.5) K /) , '#C 2/) I.*6. (.5 ->-(5& : V % ? < G G + "*$ #?*.0)% 04(/-$0-2.0 $()80-(2$D -$

HKI G HXI ,`  , HEI +Ac H;I ,

H'I W#0 U '# * @($-0-7# 1#*) 28&'#1 *2/ )#0 + U  , U  , U  , + U  , U  , U  , + U  , K *2/ G U  , U  U  , G U  , + U  + U  , G X 63 /#0 H*/Q KI > /#0H*/Q XI <+GAD 0"#2 YUZ -$ #J8*) 0(

Y'=>? 5 */Q P /#2(0#$ 0"# */Q(-20 (3 * $J8*1# &*01-? P *2/ YUZ /#2(0#$ 0"# )*19#$0 -20#9#1 )#$$ 0"*2 (1  #J8*) 0( UZ5

0X2 W#0 @ '# *2 (// @1-&# 28&'#1 *2/ ! @ '# 0"# 3())(4-29 $#0 (3 , ! 8 &'()*65-$ ! @ < K *.  '* :*D 'D. iGD+D,D555555D @ +j

012   !"# 28&'#1 (3K-2 ! @$8." 0"*0K-$ #-0"#1 $%&&#01-. (1 $U#4k$%&&#01-. (1 '(0"D*2//#0HKI /-7-$-')#  '% @ -$

HKI H@ ,%, HXI ,H@ ,% HEI H@ ,%,> + H;I ,@ ,

0112 !"# 28&'#1 (3 K -2 ! @$8." 0"*0 0"# 01*.# (3K -$ 2(0 /-7-$-')# '%@ '80 /#0 HKI -$ /-7-$-')# '% @ -$ Y'=>?5 !"# 01*.# (3 * &*01-? -$ 0"# $8& (3 -0$ /-*9(2*) #201-#$5Z

HKI H@   ,%AF,  F a ,% HXI @= AF ,%,

(23)

 <,=,>?@A*A= B C*=>@D,E   F-9G  01112  !"# 28&'#1 (3K -2! @$8." 0"*0 /#0HKI -$ 2(0 /-7-$-')# '%@ -$

HKI ,@, HXI @=  ZF HEI @=  HF H;I @=  F,

YlLL ,G+GD =>=>=>=>=Z _5++H*I W#0 P *2/ S '# 04( =   ! H #/#b-*#?=1') -<5Ib->&&5()*6 &'()*65- -=6. (.'( Wc G cW$

63 [!/#2(0#$ 0"# 01*2$@($# (3 [D 0"#2 P, S,HP! SI ,HPS ,I!-$ #J8*) 0(

HKI P, HXI c, HEI W, H;I PS

H'I W#0 P '# * = ! H &'()*+ -'(*-2>*#? W = , + G + G D P G + + < + + + *2/ P +, G G + + + 5 !"#2 0"# $8& (3 0"# /-*9(2*) #201-#$ (3 P -$ YlLL ,G++D =>=>=>b>bZ _5+,

H-I W#0 [ < Y*-QZ ' # * = ! H &'()*+ '#C 15( d G XB-QZD 4"#1# '-Q< ,->Q*-Q3(1 + -D Q =5 63 0"# /#0#1&-2*20 (3  [ -$ ,D0"#2 0"# /#0#1&-2*20 (3 0"# &*01-? _ -$

HKI ,+G HXI ,++ HEI ,+, H;I ,+=

H--I 63 [ -$ * = ! H &'()*+ -=6. (.'( e!<,[>6D4"#1#[!-$ 0"# 01*2$@($# (3 [ *2/ 6 -$ 0"# = ! H *C5#(*(>

&*01-?D 0"#2 0"#1# #?-$0$ * .()8&2 &*01-?Dq <

G G G V % ? $8." 0"*0 HKI [q < G G G

HXI [q < q HEI [q < ,q H;I [q < V

H---I 63 0"# */Q(-20 (3 * = ! H &'()*+ e *-= + + d + , b b +

D 0"#2 0"# @($$-')# 7*)8#H$I (3 0"# /#0#1&-2*20 (3[ -$H*1#I

HKI 8 HXI , HEI + H;I ,

(24)

 <,=,>?@A*A= B C*=>@D,E   F-;G  !"#$%& (%)

 "."/$01" 0 

_5+ H'I @ D,/ 0M2 bD 0W2 Aa D,8 = D,C ,Kf _5` H*I ? < + (1 ? < ,B H'I ? < b _5+G ? < G (1 ? <  ; = , , , , * ' . D,++ cG _5+, ,H *,> ', > ., > I _5+b !1-*29)# KXE -$ -$($.#)#$5 D,+6 G   D,+B d`,

 "."/$01" 00 

_5+ H*I ? < + D % < , D V < = B .(2$-$0#20 H'I ? < , D % < + D V < + B .(2$-$0#20 H.I -2.(2$-$0#20 _5, N < == , D ?: %: V < +a , : + : = _5= 8   _5b , _5d ,

_5c H*I = H'I < =D <+G H.I < =D +G

_5` ? < + > ,N D % < =N D V < ND 4"#2 @ < + B ? < ,ND % < + =N D V < N 4"#2 @ < , B 4"#1# N C  _5+G 63 N ,D +a N  , N  , + I , N  H A V = N  , % I A N  H , ? , D 63 N< ,D 0"#2 ? < D % < , , + *2/ V < G 4"#1# C  _5++H*I +`D H'I b _5+= * ' .  ' . * . * ' _5+b 63  Z (.5# + G d B % < ` d *2/ V < G B 63 < a 0"#2 ? < b a d N  B % < d ` N  += *2/ V < N 4"#1# N C  

_5+a ? < H* > ' > .I D % < *' > '. > .* D V < *'. _5+A H*I ,+B H'I aB H.I ++`

 "."/$01" 000 

_5+ ? < 2 D 2 6 _5, 63 < aD$%$0#& -$ .(2$-$0#20 4-0"-23-2-0# $()80-(2 9-7#2 '%V < ND % < + ,H=N > bI *2/ ? < + ,  HaN > ,I 4"#1# N C  63 aD $%$0#& -$ .(2$-$0#20 4-0" 82-J8# $()80-(2 9-7#2 '% V < + = H+ I B ? < + =H > , I *2/ % < G5

_5b X _5a ; _5A H*I ;B H'I K _5d =

(25)

 <,=,>?@A*A= B C*=>@D,E   F-2G 

 "."/$01":0 

_5+ +G _5, c _5= ? < , = D % < , _5b aGb` _5a r < ++ + G _5d =Hb` E HaG > +I _5c + _5` 3 H*I < +gbD * < +g,

_5++ bG+b+ +G _5+, , _5+= KX -$ 2#-0"#1 $%&&#01-. 2(1 $U#4 $%&&#01-.

_5+b b G + G = , G G + > G G G A G G a , G B b = , = = , , , + > G = = = G G = G G

_5+a H*I bD H'I c

_5+A 2 < d _5+d AaG _5+c b _5+` =g,

_5,G H*I d

H'I H-I q < , *,* +  ', ' 3(1 *D ' CB H--I q /(#$ 2(0 #?-$0B

H---I q < .* ==*. *D . C *2/ =* > . GB =' > / G5

 "."/$01":00 

_5+H*I ,,aD H'I f _5, H-I aD H--I +cD H---I c D H-7I a _5=

,a = ,+ += G +G +` b +d _5b H*I a a = d d b B H'I b, dG ,a bc +` + _5a ,` _5d ,+ _5c + _5` ?+ < +D ?, < ,J +=< + _5+G +GG _5++ + _5+, ? < ,D % < +D V < +

_5+= H-I * H J B C B H--I * < H '#C B +g= B H---I * < H J B G ,7H

_5+b H*I q< , , a= = D H ' I q < , + , +

D H.I 2( $()80-(2 _5+A c _5+d ==aa

_5+c H*I +cB H'I G _5+` HKI _B HXI TB HEI [B H;I [ _5,G HKI [D _D ! B HXI TB HEI [D C B H;I C 

 "."/$01":000 

_5+ b _5, K _5a E _5A K _5d H*I KD H'I XD H.I K

_5c HKI C HXI _DT HEI CDT H;I [DC _5` H*I K D H'I XD H.I X

References

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