matrices and determinants theory_e
TRANSCRIPT
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"aogiscbuaorpcysims.ig" 5
Znrh aotrix :O 1 RoijQa g is mo``nl o znrh aotrix, id o ij 1 6 i & j.
n.k. : (i)
666
666(ii)
666
666
666
^ppnr t riogku`o r aotr ix :O 1 RoijQa g is soil th en uppnr triogku`or, id o ij 1 6 dhr i 7 j (i.n., o`` tcn n`nangts en`hw tcn
liokhgo` n`nangts orn znrh).
n.k. : (i)
vu66
zyx6
lmeo
(ii)
z66
yx6
meo
@hwnr t riogku`or aotrix :O 1 RoijQa gis soil th en o `hwnr triogku`or aotrix, id oij1 6 dhr i 0 j. (i.n., o`` tcn n`nangts oehvn
tcn liokhgo` n`nangts orn znrh.)
n.k. : (i)
zyx
6me
66o
(ii)
6zyx
66me
666o
Liokhgo ` aotr ix :O squorn aotrix Ro ijQg is soil th en o liokhgo` aotrix id o ij1 6 dhr i j. (i.n., o` tcn n`nangts hdtcn squorn aotrix htcnr tcog liokhgo` n`nangts orn znrh)
Ghtn :Liokhgo` aotrix hd hrlnr g is lnghtnl os Liok (o??, o55, ......ogg).
n.k. : (i)
m66
6e6
66o
(ii)
m666
6666
66e6
666o
]mo`or aotrix :]mo`or aotrix is o liokhgo` aotrix ig wcimc o`` tcn liokhgo` n`nangts orn soan. O 1 RoijQgis o
smo`or aotrix, id (i) oij 1 6 dhr i j ogl (ii) oij 1 b dhr i 1 j.
n.k. : (i)
o6
6o(ii)
o66
6o6
66o
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"aogiscbuaorpcysims.ig" =
^git aot rix (ilngtity aotrix) :^git aotrix is o liokhgo` aotrix ig wcimc o`` tcn liokhgo` n`nangts orn ugity. ^git aotrix hd
hrlnr 'g' is lnghtnl eyg(hr).i.n. O 1 RoijQg is o ugit aotrix wcng o ij 1 6 dhr i j & oii 1 ?
nk. 5
1
?6
6?,
=1
?66
6?6
66?
.
Mhaporoe`n aotri mns : \wh aotrimns O & E orn soil th en mhaporoe`n, id tcny covn tcn soan hrlnr(i.n., guaenr hd rhws hd O & E orn soan ogl o`sh tcn guaenr hd mh`uags).
n.k. : (i) O 1
5?=
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"aogiscbuaorpcysims.ig"
=5
5?
, O + E 1
;4
66
?6
]uest romtihg hd aotrimns :@nt O & E en twh aotrimns hd soan hrlnr. \cng O E is lndignl os O + ( E) wcnrn E is ( ?)
E.
Vrhpnrt ins hd oll itihg smo `or au` tip`i mo tihg :Mhgsilnr o`` aotrimns hd hrlnr a g, wchsn n`nangts orn drha o snt D (D lnghtn _, U hr M).
@nt Aa g(D) lnghtn tcn snt hd o`` sumc aotrimns.
\cng
(o) O Aa g (D) & E Aa g ( D) O + E Aa g(D)(e) O + E 1 E + O
(m) (O + E) + M 1 O + (E + M)
(l) H 1 RhQa g is tcn ollitivn ilngtity.
(n) Dhr nvnry OAa g(D), O is tcn ollitivn igvnrsn.(d) (O + E) 1O +E
(k) O 1 O(c) (?+ 5) O 1?O + 5O
Au`t ip` imot ihg h d aot rimns :@nt O ogl E en twh aotrimns sumc tcot tcn guaenr hd mh`uags hd O is soan os guaenr hd rhws
hd E. i.n., O 1 Ro ijQa p & E 1 ReijQp g.
\cng OE 1 Rm ijQa g wcnrn mij 1
p
?b
bjibeo , wcimc is tcn lht prhlumt hd itc rhw vnmthr hd O ogl jtc
mh`uag vnmthr hd E.
n.k. : O 1
?=5
=5?, E 1
65??
6?66
???6
, OE 1
5;=?
?9
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"aogiscbuaorpcysims.ig" >
Vrhpnr tins h d aot rix au `ti p`imoti hg :Mhgsilnr o`` squorn aotrimns hd hrlnr 'g'. @nt Ag(D) lnghtn tcn snt hd o`` squorn aotrimns hd
hrlnr g. (wcnrn D is _, U hr M). \cng
(o) O, E A g (D) OE A g ( D)(e) Ig kngnr o` OE EO(m) (OE) M 1 O( EM)
(l) g, tcn ilngtity aotrix hd hrlnr g, is tcn au`tip`imotivn ilngtity.
Og 1 O 1gO O A g (D)(n) Dhr nvnry ghg sigku`or aotrix O (i.n., |O|6) hd Ag(D) tcnrn nxist o ugiqun (portimu`or)
aotrix EAg(D) sh tcot OE 1g1 EO. Ig tcis mosn wn soy tcot O & E orn au`tip`imotivnigvnrsn hd hgn oghtcnr. Ig ghtotihgs, wn writn E 1 O? hr O 1 E?.
(d) Id is o smo`or (O) E 1(OE) 1 O(E).(k) O(E + M) 1 OE + OM O, E, M Ag (D)(c) (O + E) M 1 OM + EM O, E, M Ag (D).
Ghtns :(?) @nt O 1 RoijQa g. \cng Og 1 O &aO 1 O, wcnrng & a orn ilngtity aotrimns hd hrlnrg & a rnspnmtivn`y.
(5) Dhr o s quorn aotrix O, O5 lnghtns OO, O= lnghtns OOO ntm.
Nxoap`n # 5 : d(x) is o quolrotim nxprnssihg sumc tcot
?mm
?ee
?oo
5
5
5
)?(d
)?(d
)6(d
1
?m5
?e5
?o5
dhr tcrnn ugnquo` guaenrs o, e, m. Digl d(x).
]h`utihg : \cn kivng aotrix nquotihg iap`ins
)?(d)?(md)6(dm
)?(d)?(ed)6(de
)?(d)?(od)6(do
5
5
5
1
?m5
?e5
?o5
x5 d(6) + xd(?) + d(?) 1 5x + ? dhr tcrnn ugnquo` guaenrs o, e, m .....(i) (i) is og ilngtity d(6) 1 6, d(?) 1 5 & d( ?) 1 ? d(x) 1 x (ox + e)
5 1 o + e & ? 1 o + e.
e 15
?& o 1
5
= d(x) 1
5
=x5 +
5
?x.
]n`d promtimn prhe`nas :
(?) Id O() 1
mhssig
sigmhs, vnridy tcot O() O() 1O(+).
Cngmn schw tcot ig tcis mosn O(). O() 1 O() . O().
(5) @nt O 1
>5?
56=
?4 ]iap`idy
oemoem
meo
meo555
]h`utihg : Kivng lntnrnaigogt is nquo` th
1oem
?
oemoemoem
meo
meo===
555
1oem
oem
???
meo
meo===
555
Opp` y M? M ? M5, M5M5 M=
1
?66
mmeeo
mmeeo=====
55555
1 (o e) (e m)
?66
mmemeeoeo
mmeeo=5555
5
1 (o e) (e m) Roe5
+ oem + om5
+ e=
+ e5
M + em5
o5
e o5
m oe5
oem e=
e5
mQ1 (o e) (e m) Rm(oe + em + mo) o(oe + em + mo)Q
1 (o e) (e m) (m o) (oe + em + mo)
Domthr \cnhrna :
^sn hd domthr tcnhrna th digl tcn vo`un hd lntnraigogt. Id ey puttigk x 1 o tcn vo`un hd o lntnraigogt
vogiscns tcng (x o) is o domthr hd tcn lntnraigogt.
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"aogiscbuaorpcysims.ig" ??
Nxoap`n # 4 Vrhvn tcot
oemoem
meo
meo555
1 (o e) (e m) (m o) (oe + em + mo) ey usigk domthr tcnhrna.
]h`utihg : @nt o 1 e
L 1oeomem
meo
meo555
1 6
Cngmn (o e) is o domthr hd lntnraigogt
]iai`or`y, `nt e 1 m, L 1 6
m 1 o, L 1 6
Cngmn, (o e) (e m) (m o) is domthr hd lntnraigogt. Eut tcn kivng lntnraigogt is hd didtc
hrlnr sh
oemoem
meo
meo
555 1 (o e) (e m) (m o) { (o5 + e5 + m5) + (oe + em + mo)}
]igmn tcis is og ilngtity sh ig hrlnr th digl tcn vo`uns hd ogl . @nto 1 6, e 1 ?, m 1 ?
5 1 (5) (5 )(5 ) 1 ?. ........(i)@nt o 1 ?, e 1 5, m 1 6
566
6
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"aogiscbuaorpcysims.ig" ?5
(4) ]iap`idy5
55
5
ooeomomem
oeeeoooe
omemmeoee
.
(;) Vrhvn tcoteomm5m5
e5omee5
o5o5meo
1 (o + e + m)=.
(3) ]chw tcot
oem?
moe?
emo?
1 (o e) (e m) (m o) ey usigk domthr tcnhrna .
Ogswnrs : (>) 6 (4) 6
Au`t ip` imo tihg h d twh lntnraigogts :Id O ogl E orn twh squorn aotrimns hd soan hrlnr, tcng |OE| 1 |O| |E|.
55?555?5
5???5???
55
??
55
??
aeaoeoaeaoeo
aa
eoeo
===
555
???
meo
meo
meo
===
555
???
ga
ga
ga
1
==5=?===5=?===5=?=
=555?5=555?5=555?5
=?5???=?5???=?5???
gmgegoamaeaomeo
gmgegoamaeaomeo
gmgegoamaeaomeo
Ghtn :Os |O| 1 |O|, wn covn |O| |E| 1 |OE| (rhw - rhw antchl)|O| |E| 1 |OE| (mh`uag - mh`uag antchl)
|O| |E| 1 |OE| (mh`uag - rhw antchl)
Nxoap`n # ; Digl tcn vo`un hd=?
5?
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"aogiscbuaorpcysims.ig" ?=
]h`utihg. Kivng lntnraigogt mog en sp`ittnl igth prhlumt hd twh lntnraigogts
i.n.
====5=5=?=?=
=5=55555?5?5
=?=?5?5?????
yexoyexoyexo
yexoyexoyexo
yexoyexoyexo
1
===
555
???
meo
meo
meo
666
yyy
xxx
=5?
=5?
1 6
Nxoap`n # 9 Vrhvn tcot5
==5
5=5
?=
5=5
555
5?5
5=?55?5??
)eo()eo()eo(
)eo()eo()eo()eo()eo()eo(
1 5(o? o
5) (o
5 o
=) (o
= o
?) (e
? e
5) (e
5 e
=) (e
= e
?).
]h`utihg.
5==
55=
5?=
5=5
555
5?5
5=?
55?
5??
)eo()eo()eo(
)eo()eo()eo(
)eo()eo()eo(
1
==5
=5
=5=5
55
=?=5
?5
=
=55
=5
5555
55
5?55
?5
5
=?
5
=
5
?5?
5
5
5
???
5
?
5
?
eo5eoeo5eoeo5eo
eo5eoeo5eoeo5eoeo5eoeo5eoeo5eo
1
=5
=
55
5
?5
?
o5?o
o5?o
o5?o
=5?
5=
55
5?
eee
eee
???
1 5
=5
=
55
5
?5
?
oo?
oo?
oo?
=5
=
55
5
?5
?
ee?
ee?
ee?
1 5(o? o
5) (o
5 o
=) (o
= o
?) (e
? e
5) (e
5 e
=) (e
= e
?)
Ghtn : \cn oehvn prhe`na mog o`sh en sh`vnl usigk domthr tcnhrna antchl.
]n`d promtimn prhe`nas
(9) Digl tcn v o`un hd
555
555
555
moe5oe
oemo5m
emoem5
(?6) Id O, E, M orn rno` guaenrs tcng digl tcn vo`un hd 1?)MEmhs()MOmhs(
)EMmhs(?)EOmhs(
)OMmhs()OEmhs(?
.
Ogswnrs : (9) (=oem o= e= m=)5 (?6) 6
]uaaotihg h d lntnraigogts : @nt (r) 1=5?
=5?
eee
ooo
)r(c)r(kd(r)
wcnrn o?, o
5, o
=, e
?, e
5, e
=orn mhgstogts
iglnpnlngt hd r, tcng
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"aogiscbuaorpcysims.ig" ?
1
6?5
??=
g?5?55 5g5g5g
M? M
? 5 M
5
1
6?6
???
g?5?5555 5g5g?g5g
1 (?)??
g?5555 5g?g5g
1 5g ? g =
Nxoap`n # ?5 Idr1
5??r
r=r5
6??r
, digl
g
?r
r
]h`utihg. Hg nxpogsihg hd lntnraigngt, wn knt
Lr1 (r ?) (= r) + ; + r5 +
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"aogiscbuaorpcysims.ig" ?4
Nxoap`n # ?= Id d(x) 15
Vrhvn tcot D lnpngls hg`y hg x?, x
5ogl x
=
D 1
5=?5=55?
555??
5?
?=?5??
exexexexexex
oxoxox
???
ogl siap`idy D.
]h`utihg :?lo
lD1
5=?5=55?
555??
5?
?=?5??
exexexexexex
oxoxox
666
+
5=?5=55?
555??
5? exexexexexex
???
???
+
666
oxoxox
???
?=?5?? 1 6
Cngmn D is iglnpnglngt hd o?.
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"aogiscbuaorpcysims.ig" ?;
]iai`or`y?le
lD1
5le
lD1 6.
Cngmn D is iglnpnglngt hd e?ogl e
5o`sh.
]h D is lnpnglngt hg`y hg x?, x
5, x
=
Vut o?
1 6, e?
1 6, e5
1 6
D 15=
55
5?
=5?
xxx
xxx
???
1 (x? x
5) (x
5 x
=) (x
= x
?).
Nxoap`n # ?4 Id)x?(gxmhs
xsignx
1 O + Ex + Mx5 + ....., tcng digl tcn vo`un hd O ogl E.
]h`utihg : Vut x 1 6 ig
)x?(gxmhs
xsign x
1 O + Ex + Mx5 + .......
6?
6?1 OO
O 1 6.
Liddnrngtiotigk tcn kivng lntnraigogt w.r.t x, wn knt
)x?(gxmhs
xmhsnx
+
x?
?xsig
xsignx
1 E + 5 M x + ......
Vut x 1 6, wn knt
6?
??
+ ?6
6?
1 6
E 1 ? + ? 1 6 O 1 6, E 1 6
]n`d promtimn prhe`na
(?5) Id x??x
??xx5
x?xx
1 ox= + ex5 + mx + l. Digl
(i) l (ii) o + e + m + l (iii) e
Ogswnrs : (?5) (i) ? (ii) > (iii)
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"aogiscbuaorpcysims.ig" 5?
@nt z 1 t?, y 1 t
5 x 1 5 t
? t
5
wcnrn t?, t
5 U.
Nxoap`n # 55 Mhgsilnr tcn dh``hwigk systna hd nquotihgs
x + y + z 1 4
x + 5y + =z 1 ?6
x + 5y + z 1Digl vo`uns hdoglid sumc tcot snts hd nquotihg covn(i) ugiqun sh`utihg (ii) igdigitn sh`utihg
(iii) gh s h` ut ihg
]h`utihg : x + y + z 1 4
x + 5y + =z 1 ?6
x + 5y + z 1
L 15?
=5?
???
Cnrn dhr 1 = snmhgl ogl tcirl rhws orn ilngtimo` cngmn L 1 6 dhr 1 =.
L?1
5
=5?6
??4
L51
?
=?6?
?4?
L=1
5?
?65?
4??
Id 1 = tcng L?1 L
51 L
= 1 6 d hr 1 ?6
(i) Dhr ugiqun s h`ut ihg L6i.n. =
(ii) Dhr igdigitn sh`utihgs
L 1 6 1 =L
?1 L
51 L
=1 6 1 ?6.
(iii) Dhr gh sh`utihgL 1 6 1 =Ot`nost hgn hd L
?, L
5hr L
=is ghg znrh ?6.
]n`d promtimn prhe`nas
(*?=) ]h`vn tcn dh``hwigk systna hd nquotihgs
x + 5y + =z 1 ?
5x + =y +
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"aogiscbuaorpcysims.ig" 55
x + 5y + =z 1 6
5x + =y + ) ]h`vn: (e + m) (y + z) ox 1 e m, (m + o) (z + x)ey 1 m o, (o + e) (x + y) mz 1 o ewcnrn o + e + m6.
(?4) @nt 5x + =y + < 1 6 2 =x + >y + 4 1 6, 5x5 + 4xy + >y5 + 3x + ?5y + ? + t 1 6, id tcn systna hd
nquotihgs ig x ogl y orn mhgsistngt tcng digl tcn vo`un hd t.
Ogswnrs : (?=) x 1 ? + t y 1 5t z 1 t wcnrn t U(?) x 1 m e
o e m
, y 1 o m
o e m
, z 1 e o
o e m
(?4) t 1 ;
Opp`imotihg hd lntnraigogts : Dh``hwigk nxoap`ns hd schrt cogl writigk `orkn nxprnssihgs orn:(i) Orno hd o triogk`n wchsn vnrtimns orn (x
r, y
r)2 r 1 ?, 5, = is:
L 1?
5 ?yx
?yx
?yx
==
55
??
Id L 1 6 tcng tcn tcrnn phigts orn mh``ignor.
(ii) Nquotihg hd o stroikct `ign possigk tcrhukc (x?, y
?) & (x
5,y
5) is
?yx
?yx
?yx
55
?? 1 6
(iii) \cn igns: o?x + e?y + m? 1 6........ (?)o
5x + e
5y + m
51 6........ (5)
o=x + e
=y + m
=1 6........ (=)
orn mhgmurrngt id,
===
555
???
meo
meo
meo
1 6.
Mhglitihg dhr tcn mhgsistngmy hd tcrnn siau`tognhus `ignor nquotihgs ig 5 vorioe`ns.
(iv) ox + 5 cxy + ey + 5kx + 5 dy + m 1 6 rnprnsngts o poir hd stroikct `igns id:
oem + 5dkc od ek mc 1 6 1mdkdec
kco
]igku`or ghg sigku`or aotrix : O squorn aotrix O is soil th en sigku`or hr ghg- sigku`or ommhrligkos |O| is znrh hr ghg-znrh rnspnmtivn`y.
Mhdom thr aot rix oljh igt aot rix : @nt O 1 RoijQg en o squorn aotrix. \cn aotrix hetoignl eyrnp`omigk nomc n`nangt hd O ey mhrrnsphgligk mhdomthr is mo``nl os
mhdomthr aotrix hd O, lnghtnl os mhdomthr O. \cn trogsphsn hd mhdomthr
aotrix hd O is mo``nl os oljhigt hd O, lnghtnl os olj O.
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"aogiscbuaorpcysims.ig" 5=
i.n. id O 1 RoijQgtcng mhdomthr O 1 RmijQg wcng mij is tcn mhdomthr hd o ij i & j.Olj O 1 Rl ijQgwcnrn lij 1 mji i & j.
Vrhpnrtins hd mhdomthr O ogl olj O:(o) O . olj O 1 |O|g1 (olj O) O wcnrn O 1 RoijQg.(e) |olj O| 1 |O|g ?, wcnrn g is hrlnr hd O.
Ig portimu`or, dhr = = aotrix, |olj O| 1 |O|5
(m) Id O is o syaantrim aotrix, tcng olj O orn o`sh syaantrim
aotrimns.
(l) Id O is sigku`or, tcng olj O is o`sh sigku`or.
Nxoap`n # 5= : Dhr o == sbnw-syaantrim aotrix O, schw tcot olj O is o syaantrim aotrix.
]h`utihg : O 1
6me
m6o
eo6
mhd O 1
5
5
5
ooemo
oeeem
moemm
olj O 1 (mhd O) 1
5
5
5
ooemo
oeeem
moemm
wcimc is syaantrim.
Igvnrsn hd o aotrix (rnmiprhmo` aotrix) :@nt O en o ghg-sigku`or aotrix. \cng tcn aotrix
|O|
?olj O is tcn
au`tip`imotivn igvnrsn hd O (wn mo`` it igvnrsn hd O) ogl is lnghtnl ey O?.[n covn O (olj O) 1 |O|g1 (olj O) O
O
Oolj
|O|
?1g1
Oolj
|O|
?O, dhr O is ghg-sigku`or
O? 1|O|
?olj O.
U n a o r b s :
?. \cn gnmnssory ogl suddimingt mhglitihg dhr nxistngmn hd igvnrsn hd O is tcot O is ghg-sigku`or.
5. O? is o`woys ghg-sigku`or.
=. Id O 1 lio (o??, o55, ....., ogg) wcnr n oii6i, tcng O? 1 liok (o??
?, o55?, ...., ogg
?).
. (O?)? 1 O id O is ghg-sigku`or.
4. @nt b en o ghg-znrh smo`or & O en o ghg-sigku`or aotrix. \cng (bO)? 1b
?O?.
;. |O?| 1|O|
? dhr |O| 6.
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"aogiscbuaorpcysims.ig" 5
(=) Id O is sigku`or ogl (olj O) E 6, tcng tcn systna cos gh sh`utihg(wn soy it is igmhgsistngt).
Chahkngnhus systna ogl aotrix igvnrsn :Id tcn oehvn systna is chahkngnhus, g nquotihgs ig g ugbghwgs, tcng ig tcn aotrix dhra it is OS 1 H.
( ig tcis mosn e? 1 e5 1 ....... eg 1 6), wcnrn O is o squorn aotrix.
Unsu` ts : (?) Id O is ghg-sigku`or, tcn systna cos hg`y tcn trivio` sh`utihg (znrh sh`utihg) S 1 6(5) Id O is sigku`or, tcng tcn systna cos igdigitn`y aogy sh`utihgs (igm`uligk tcn trivio`
sh`utihg) ogl cngmn it cos ghg-trivio` sh`utihgs.
Uogb hd o aotrix :@nt O 1 Ro ijQag. O goturo` guaenr is soil th en tcn rogb hd O id O cos o ghg-sigku`orsueaotrix hd hrlnr ogl it cos gh ghg-sigku`or sueaotrix hd hrlnr ahrn tcog. Uogbhd znrh aotrix is rnkorlnl th en znrh.
nk . O 1
6>666566
>5?=
wn covn
56
5=os o ghg-sigku`or sueaotrix.
\cn squorn aotrimns hd hrlnr = orn
>66
566
5?=
,
666
666
>?=
,
6>6
656
>5=
,
6>6
656
>5?
ogl o`` tcnsn orn sigku`or. Cngmn rogb hd O is 5.
N`nangto ry rhw trogsdhraotihg h d aotrix :\cn dh``hwigk hpnrotihgs hg o aotrix orn mo``nl os n`nangtory rhw trogsdhraotihgs.
(o) Igtnrmcogkigk twh rhws.
(e) Au`tip`imotihgs hd o`` tcn n`nangts hd rhw ey o ghgznrh smo`or.
(m) Ollitihg hd mhgstogt au`tip`n hd o rhw th oghtcnr rhw.
Ghtn : ]iai`or th oehvn wn covn n`nangtory mh`uag trogsdhraotihgs o`sh.
Unaorbs :?. N`nangtory trogsdhraotihg hd o aotrix lhns ght oddnmt its rogb.
5. \wh aotrimns O & E orn soil th en nquivo`ngt id hgn is hetoignl drha htcnr usigk n`nangtory
trogsdhraotihgs. [n writn OE.
Nmcn`hg dhra hd o aotrix : O aotrim is soil th en ig Nmcn`hg dhra id it sotisdy tcn dh``hwigks:(o) \cn dirst ghg-znrh n`nangt ig nomc rhw is ? & o`` tcn htcnr n`nangts ig
tcn mhrrnsphgligk mh`uag (i.n. tcn mh`uag wcnrn ? oppnors) orn znrhns.
(e) \cn guaenr hd znrhns endhrn tcn dirst ghg znrh n`nangt ig ogy ghg znrh
rhw is ght ahrn tcog tcn guaenr hd sumc znrhns ig summnnligk rhws.
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"aogiscbuaorpcysims.ig" 54
Unsu`t : Uogb hd o aotrix ig Nmcn`hg dhra is tcn guaenr hd ghg znrh rhws (i.n. guaenr hd rhws witcot`nost hgn ghg znrh n`nangt.)
Unaorb : \h digl tcn rogb hd o kivng aotrix wn aoy rnlumn it th Nmcn`hg dhra usigk n`nangtory rhwtrogsdhraotihgs ogl tcng mhugt tcn guaenr hd ghg znrh rhws.
]ystna hd `ignor nquotihgs rogb hd aotrix :@nt tcn systna en OS 1 E wcnrn O is og a g aotrix, S is tcn g-mh`uag vnmthr & E is tcn a-mh`uag
vnmthr. @nt ROEQ lnghtn tcnoukangtnl aotrix (i.n. aotrix hetoignl ey ommnptigk n`nangts hd E os g
+ ?tc mh`uag & dirst g mh`uags orn tcot hd O). (O) lnghtn rogb hd O ogl (ROEQ) lnghtn rogb hd tcnoukangtnl aotrix.
M`nor`y(O) (ROEQ).
Unsu` ts : (?) Id (O) 0(ROEQ) tcng tcn systna cos gh sh`utihg (i.n. systna is igmhgsistngt).(5) Id (O) 1(ROEQ) 1 guaenr hd ugbghwgs, tcng tcn systna cos ugiqun sh`utihg.
(ogl cngmn is mhgsistngt)
(=) Id (O) 1(ROEQ) 0 guaenr hd ugbghwgs, tcng tcn systnas cos igdigitn`y aogy sh`utihgs
(ogl sh is mhgsistngt).
Chahkngnhus systna rogb hd aotrix :@nt tcn chahknghus systna en OS 1 6, a nquotihgs ig 'g' ugbghwgs. Ig tcis mosn E 1 6 ogl sh (O)1(ROEQ). Cngmn id(O) 1 g, tcng tcn systna cos hg`y tcn trivio` sh`utihg. Id(O) 0 g, tcng tcn systnacos igdigitn`y aogy sh`utihgs.
Nxoap`n # 5> : ]h`vn tcn systna
?zyx5
5zyx
4zyx
usigk aotrix igvnrsn.
]h`utihg : @nt O 1
??5
???
???
, S 1
z
y
x
& E 1
?
5
4
.
\cng tcn systna is OS 1 E.
|O| 1 4. Cngmn O is ghg sigku`or.
Mhdomthr O 1
565
?=5
==6
olj O 1
5?=
6==
556
O? 1|O|
?olj O 1
4
?
5?=
6==
556
1
=/?4/?5/?
65/?5/?
=/?=/?6
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"aogiscbuaorpcysims.ig" 5;
S 1 O? E 1
=/?4/?5/?
65/?5/?
=/?=/?6
?
5
4
i.n.
z
y
x
1
=
5
?
x 1 ?, y 1 5, z 1 =.
Nxoap`n # 54 : \nst tcn mhgsistngmy hd tcn systna
3.z
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"aogiscbuaorpcysims.ig" 53
]n`d promtimn prhe`nas:
(5?) O 1
??=
=5?
5?6
. Digl tcn igvnrsn hd O usigk |O| ogl olj O.
(55) Digl rno` vo`uns hd ogl sh tcot tcn dh``hwigk systnas cos
(i) ugiqun sh`utihg (ii) igdigitn`y aogy sh`utihgs (iii) Gh sh`utihg.x + y + z 1 4
x + 5y + =z 1 ?
x + 5y +z 1
(5=) Diglsh tcot tcn dh``hwigk chahkngnhus systna covn o ghg znrh sh`utihgx + 5y + =z 1x=x + y + 5z 1 y5x + =y + z 1z
Ogswnrs : (5?)
5
??
5
?5
==
5
?5
>O? 1 OO5 + O >.
]h`utihg : [n covn tcn mcoromtnristim nquotihg hd O.| O x | 1 6
i.n.
x?66
6x?5
65x?
1 6
i.n. x= + x5 >x > 1 6.^sigk Moy`ny - Coai`thg tcnhrna.
O= + O5 >O > 1 6 > 1 O= + O5 >OAu`tip`yigk ey O?, wn knt
>O? 1 O5 + O >
Gi`phtngt aotr ix :O squorn aotrix O is soil th en gi`phtngt ( hd hrlnr 5) id, O5 1 H. O squorn aotrix is soil th en gi`phtngt
hd hrlnr p, id p is tcn `nost phsitivn igtnknr sumc tcot Op 1 H.
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Ilnaphtng t aot rix :O squorn aotrix O is soil th en ilna phtngt id, O5 1 O.
n.k.
?6
6?is og ilnaphtngt aotrix.
Ig vh `u th ry aot rix :O squorn aotrix O is soil th en igvh`uthry id O5 1, enigk tcn ilngtity aotrix.
n.k. O 1
?6
6?is og igvh`uthry aotrix.
Hrtchkhgo` aot rix :O s quor n aotrix O is soil th en og hrtchkhgo` aot rix id,
OO 1 1 OO.
Nxoap`n # 53 : ]chw tcot o squorn aotrix O is igvh`uthry, idd ( O) ( + O) 1 6
]h`utihg : @nt O en igvh`uthry
\cng O5 1( O) (+ O) 1 +O O O5
1 + O O O51 O5
1 6Mhgvnrs`y, `nt ( O) (+ O) 1 6 +O O O5 1 6 + O O O 5 1 6 O5 1 6 O is igvh`uthry
]n`d promtimn prhe`nas
(5) Id O is o gi`phtngt aotrix hd iglnx 5, schw tcot O ( + O)g 1 O dhr o`` g G.
(54) O is o sbnw syaantrim aotrix, sumc tcot O5 + 1 6. ]chw tcot O is hrtchkhgo` ogl is hd nvnghrlnr.
(5;) @nt O 1
6oe
o6m
em6
. Id OO= +O 1 6, digl .
Ogswnr (