Matrices and Determinants Theory_E

8/12/2019 Matrices and Determinants Theory_E

1/29


2/29

AO\C]

"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 riogkuò r aotr ix :O 1 RoijQa g is soil th en uppnr triogkuòr, 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 riogkuòr aotrix :O 1 RoijQa gis soil th en o `hwnr triogkuòr 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

]moòr aotrix :]moòr aotrix is o liokhgo` aotrix ig wcimc o`` tcn liokhgo` n`nangts orn soan. O 1 RoijQgis o

smoòr 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


3/29

AO\C]

"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?=


4/29

AO\C]

"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 òr au` tipì 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 mhùags 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

mhùag vnmthr hd E.

n.k. : O 1

?=5

=5?, E 1

65??

6?66

???6

, OE 1

5;=?

?9


5/29

AO\C]

"aogiscbuaorpcysims.ig" >

Vrhpnr tins h d aot rix au `ti pìmoti 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`tipìmotivn ilngtity.

Og 1 O 1gO O A g (D)(n) Dhr nvnry ghg sigkuòr aotrix O (i.n., |O|6) hd Ag(D) tcnrn nxist o ugiqun (portimuòr)

aotrix EAg(D) sh tcot OE 1g1 EO. Ig tcis mosn wn soy tcot O & E orn au`tipìmotivnigvnrsn hd hgn oghtcnr. Ig ghtotihgs, wn writn E 1 O? hr O 1 E?.

(d) Id is o smoòr (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).

]hùtihg : \cn kivng aotrix nquotihg iapìns

)?(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 ]iapìdy

oemoem

meo

meo555

]hùtihg : 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 voùn hd lntnraigogt. Id ey puttigk x 1 o tcn voùn hd o lntnraigogt

vogiscns tcng (x o) is o domthr hd tcn lntnraigogt.


11/29

AO\C]

"aogiscbuaorpcysims.ig" ??

Nxoap`n # 4 Vrhvn tcot

oemoem

meo

meo555

1 (o e) (e m) (m o) (oe + em + mo) ey usigk domthr tcnhrna.

]hùtihg : @nt o 1 e

L 1oeomem

meo

meo555

1 6

Cngmn (o e) is o domthr hd lntnraigogt

]iaiòr`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 voùns 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


12/29

AO\C]

"aogiscbuaorpcysims.ig" ?5

(4) ]iapìdy5

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| (mhùag - mhùag antchl)

|O| |E| 1 |OE| (mhùag - rhw antchl)

Nxoap`n # ; Digl tcn voùn hd=?

5?


13/29

AO\C]

"aogiscbuaorpcysims.ig" ?=

]hùtihg. Kivng lntnraigogt mog en spìttnl 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

?).

]hùtihg.

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 oùn hd

555

555

555

moe5oe

oemo5m

emoem5

(?6) Id O, E, M orn rno` guaenrs tcng digl tcn voùn 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


14/29

AO\C]

"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 +


16/29

AO\C]

"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?.


17/29

AO\C]

"aogiscbuaorpcysims.ig" ?;

]iaiòr`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 voùn hd O ogl E.

]hùtihg : 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)


21/29

AO\C]

"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 voùns hdoglid sumc tcot snts hd nquotihg covn(i) ugiqun shùtihg (ii) igdigitn shùtihg

(iii) gh s h` ut ihg

]hùtihg : 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 hùt ihg L6i.n. =

(ii) Dhr igdigitn shùtihgs

L 1 6 1 =L

?1 L

51 L

=1 6 1 ?6.

(iii) Dhr gh shùtihgL 1 6 1 =Ot`nost hgn hd L

?, L

5hr L

=is ghg znrh ?6.


(*?=) ]h`vn tcn dh``hwigk systna hd nquotihgs

x + 5y + =z 1 ?

5x + =y +


22/29

AO\C]


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 voùn 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 ;

Oppìmotihg hd lntnraigogts : Dh``hwigk nxoap`ns hd schrt cogl writigk òrkn 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`ìgnor.

(ii) Nquotihg hd o stroikct ìgn 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 ìgnor nquotihgs ig 5 vorioe`ns.

(iv) ox + 5 cxy + ey + 5kx + 5 dy + m 1 6 rnprnsngts o poir hd stroikct ìgns id:

oem + 5dkc od ek mc 1 6 1mdkdec

kco

]igkuòr ghg sigkuòr aotrix : O squorn aotrix O is soil th en sigkuòr hr ghg- sigkuòr 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 eyrnpòmigk 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.


23/29

AO\C]

"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 portimuòr, 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 sigkuòr, tcng olj O is o`sh sigkuòr.

Nxoap`n # 5= : Dhr o == sbnw-syaantrim aotrix O, schw tcot olj O is o syaantrim aotrix.

]hùtihg : 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-sigkuòr aotrix. \cng tcn aotrix

|O|

?olj O is tcn

au`tipìmotivn 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-sigkuòr

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-sigkuòr.

5. O? is o`woys ghg-sigkuòr.

=. Id O 1 lio (o??, o55, ....., ogg) wcnr n oii6i, tcng O? 1 liok (o??

?, o55?, ...., ogg

?).

. (O?)? 1 O id O is ghg-sigkuòr.

4. @nt b en o ghg-znrh smoòr & O en o ghg-sigkuòr aotrix. \cng (bO)? 1b

?O?.

;. |O?| 1|O|

? dhr |O| 6.


24/29

AO\C]


(=) Id O is sigkuòr ogl (olj O) E 6, tcng tcn systna cos gh shùtihg(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-sigkuòr, tcn systna cos hg`y tcn trivio` shùtihg (znrh shùtihg) S 1 6(5) Id O is sigkuòr, tcng tcn systna cos igdigitn`y aogy shùtihgs (igmùligk tcn trivio`

shùtihg) ogl cngmn it cos ghg-trivio` shùtihgs.

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-sigkuòrsueaotrix hd hrlnr ogl it cos gh ghg-sigkuòr 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-sigkuòr sueaotrix.

\cn squorn aotrimns hd hrlnr = orn

>66

566

5?=

,

666

666

>?=

,

6>6

656

>5=

,

6>6

656

>5?

ogl o`` tcnsn orn sigkuòr. 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`tipìmotihgs hd o`` tcn n`nangts hd rhw ey o ghgznrh smoòr.

(m) Ollitihg hd mhgstogt au`tip`n hd o rhw th oghtcnr rhw.

Ghtn : ]iaiòr th oehvn wn covn n`nangtory mhùag 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 mhùag (i.n. tcn mhùag 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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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 ìgnor nquotihgs rogb hd aotrix :@nt tcn systna en OS 1 E wcnrn O is og a g aotrix, S is tcn g-mhùag vnmthr & E is tcn a-mhùag

vnmthr. @nt ROEQ lnghtn tcnoukangtnl aotrix (i.n. aotrix hetoignl ey ommnptigk n`nangts hd E os g

+ ?tc mhùag & dirst g mhùags 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 shùtihg (i.n. systna is igmhgsistngt).(5) Id (O) 1(ROEQ) 1 guaenr hd ugbghwgs, tcng tcn systna cos ugiqun shùtihg.

(ogl cngmn is mhgsistngt)

(=) Id (O) 1(ROEQ) 0 guaenr hd ugbghwgs, tcng tcn systnas cos igdigitn`y aogy shùtihgs

(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` shùtihg. Id(O) 0 g, tcng tcn systnacos igdigitn`y aogy shùtihgs.

Nxoap`n # 5> : ]h`vn tcn systna

?zyx5

5zyx

4zyx

usigk aotrix igvnrsn.

]hùtihg : @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 sigkuòr.

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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]n`d promtimn prhe`nas:

(5?) O 1

??=

=5?

5?6

. Digl tcn igvnrsn hd O usigk |O| ogl olj O.

(55) Digl rno` voùns hd ogl sh tcot tcn dh``hwigk systnas cos

(i) ugiqun shùtihg (ii) igdigitn`y aogy shùtihgs (iii) Gh shùtihg.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 shùtihgx + 5y + =z 1x=x + y + 5z 1 y5x + =y + z 1z

Ogswnrs : (5?)

5

??

5

?5

==

5

?5

>O? 1 OO5 + O >.

]hùtihg : [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 ù th ry aot rix :O squorn aotrix O is soil th en igvhùthry id O5 1, enigk tcn ilngtity aotrix.

n.k. O 1

?6

6?is og igvhùthry 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 igvhùthry, idd ( O) ( + O) 1 6

]hùtihg : @nt O en igvhùthry

\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 igvhùthry


(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 (

Matrices and Determinants Theory_E

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