Chapitre 3-Les matrices algèbre 02

L er 'r,',Jr-.g'r.

Q^ ruil,,0.r U(= K s{^ q- '

p c,rnnLç"I;

t"'n) S=\L ïn,h G ulv4. U^{ -*h*co A J"5* t^^L&u" veJrcu*1u!*" \ir - \ <nh*^ -v cJ*-, zrrrv*t*t I - ,- )' ;. .-"f{\d.ù^h, I ,

C*

I

,\

o,h ^ .^r|.n** w\ Q,,..,,rr.,a

J'd{o'**ul,-, .ta-K.

t ^ b..'Qt-'*t{n

0

*!* {* f *O* e^"'r*Æ "hÀ'(1^^l-(^1-.'

Jq'^* K ' *r{- J^ p. *.0o*ne^,

8- .*{lida^lâ Jo {* ^'^'h cn14' 0* ^Je

A = ta"i) 4-.\s1a,1 ^sûs\^

' lJ" Q*."{'hv\€ Ë*v*yl& "

A=

u^Aq^.*ht .'e

_..-...

J" \rG" d\

[a ,r) ot'ra(/ \

'*, à^t It J^ * J^ '4- * L N,t "Jt.!n 9f K\ o\*- \K *Ï *J* H^,^C

Q,.l= 4

(^,^) À *{-T,.iu,L

^-/ nXS {r5t"1**'.L["o.oJ' don *ttr olorÆ a^ 41t .j {^l; c.,s\^,(,.1,-;\4 "*h

^,

I Ue,tt:< *^,bù*

.

t

.t [à=txi) e Hr,^,(,].)

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ffi=*l , n:h\"h

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ô q. .'"-t{iciab ^*t" l l

€ Lo. *o,ln^* -L h |

. . n

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#uo, *Â o1ï "l.o^,

(r, H) dtf 6 U^( mc^,[,r,"æ,J*\f" ?^*tlr

^*hi* 0,.fo "CI^

À = [ * , ^ n e ^ a . ' 'o ^ r )

. uAr *.^[ù* & g\*

c t*ï/u-

Ja

û\ z ^

'0, *iui* .JsvrYr* twrr^) "J "TrJ"" *^^

-fu {.'l* ( uné** qgr",nhv*

N*.[ *) ft ^ C,K) *^ 0^* J*

'L% éU^rAÂ a^h,s*,- .,own (f J'/ Qo-^*,il* A . t$" ?* W ct

a,,^o-t* *{.o ^n-),

f@"\- i,ào\,t,-,vr\ rJîno i+;- )o."tJiro. .y^" A r'rv çi J"

û , t n r . u r n a * ùJùr4- . 1 c n n * û

cln J.,iI o.

^ ^h

J * Q**Ja a^:

{

P!^

qïi=4'

vt^*"[,^.***tr' "

Vr'e\4t-tn\-

Cea^{o *"I^*

3^=

h*nl-i'>J l'T Av' .*{( ,, atb u,à ^*h ^*t^ f-'gr^'\ I

a

.

{ , t g . À lL- < )

r

J"-Lhti

ofu f*

\

).

M^e**b^* .h.ft"û"

uËi;q* u^ *oJL* h*.*1^l,an<

aff-u^,*t\"' * h"'*y^tc*'rr t1tdo*

.}.'*^{ ^ -^)r^.* c,o^..,6. . k q^^eto^J,^ct co,rr-,.rJtutafq H , È= [o,;) 'Q.^ P*S *,ho.. . ç.il,,À [), Q-. c^fnî{ -q, {* .uLl.-;1,,^*Q-"

îr ^ tr

_

,_,eW

r.[t-i)=.r+] =4"

r vn ci)çr- tJI. . "lt-i.*.

*t

\ o"uu*.-p

'àrl.--q^^,f p-(a;y) ,* h=|tr,) Jq^* r,n*/"1.a ".1*,^{*"Tf . \*,*r,or<rr,nnua A+ û *,t- {^ vn*h;"* *1 [r^nrnJ

l*,

f

t*,^) "fufi'rir

*44' cte',r/b ilcl^-

clnrnsrrvr,rrr-e "--|ôa*lf

-i

hg,-"É, ç,' A ; (i

) &t' B=(i!)

"J*

-l) -ïli)=(:, A+ o= li î)-\i^\)=rJl, '(

( .

Pn J-T ",

J'o^l

Pc"Jc'iJ4 '

c lr1^

Jl ,^l *^lo^ *

l-* t, .' .I*t

5 = tul, ) =l' H*nt*)

pcJ'";+a ç'^ wn

n) J/ti *iq{'h'" ['' tlTu -b .o .k n.rtd, 8* A a vr^",.h,.' , d.e e^,1. ' c(A= 1'fol;) o ' -1 \ 5 .^-QrE"\- d,a? t:^4*^,--(à'^) ç' h = f /

or

f \

\

a.

^

ù

)

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tP,

: \

'*'.J"UU"..ln .k { .L .'V*,i Pn,r'".^," ",k

*,'$,hi

: t c,".h'../h

{tt'"'rt'

Wvrtr,CoJc*-/ €

t{"ÇLAi",

4,f ev'çstc-{-A,Ê,Cel(*,,Ck)-,t"Àr{.

cfrne^ Qt Xn-y^Jbl

a- vo^|.e :

A+{à= B*A

,t)

,{,) G*D^C

' Qonnsr\nnrr-c "*

= At tÊ*c) "' Q"^ârsryrnrrua*h"-ot't*h*e'

A= = go*"a*1 À + c$*n(*\

n)

csm/v.tb*hw"

P*.^"h"uu^J'L "n*.,-k)

-h {'* J h".{^t-{'dlo'^,*l"U "en^JuC q)

A+ [-n)=C'A)*A=

A'-'fu^^tlfih1* 5)

Oï,*,,nqL\

-l* A , \tu

c { \ . 4 + Ô ) = o (A + q È

6 ) [ a * p À = " ( - â +P h ")

r)

klds?

L,È + , ,) lte^o<^b-4J^* Jùt'* "[Jûh!" .1" Jq^ x. ^ *Ï* u^ *] ^

-L Q* ^.1h1À.J'.n

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f?.0'*à ;.*h , ^ t p,t^,i

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u) *-

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T{""ul* nn-{...

4.*,.nJ*'r*r,

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Qe^**,\i' 1-s- t;; , i=4t-'hr i='t,-n

p^-.h6-,' -["wk Q'"*-{rtr'ur,;-r{. J^cw* i , t*^l-, ar.f'* ; "F (

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(

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v^'|v 1",*<

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'= hn"n. il*.^( k )

.^I yf&i

/l

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71rafsrn

L^ \^n< c.rv.o*i "1^q

t ( ^, o\ ë[ o l\ q.[ o)'o'\ ]%\ o ù /,=\o 1

l-

n n "f)' t . r I n J*,-n" h.h t+ "^] "t-o-t.'ili, m.zlf

trr{o .1" F *,t gtrJ 0

t,rçnh"* c-rr,t

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t,à

Liro

i . \ 4 , - , t ^ \ c l 1 e 1a - , 1 \ '

tni = QioLn;* 'q*nboj ,

r

lt

[\ Jr t

b x!,wt çr{-t

A,Se \*,nr(*)

Irnn^a,ut, q,l\ r* 1

A .f4=

L,,*f À t* ttÀ.'t C..*^rJdrq,

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h,{4= t i :) ("

a^ t.r"s,b ,onJ^*.

6ar^v*"[a .

#= (t^ ^-) ' ô&^n "-)-u

(,-l =fli: ;) )= ::: î)

\fi

b A l=i ; ) t i i) A Ê ={c

o C0^.. A - , \t

h.(à=

o ù

r 1 ) ro+ i )( : -7

ù

À-_0 (4-

f'.

f o ,

t -

lov\ t^$hhe

R . t nc v r c w ^ . ( ) "

Ê"=-C

h A ^ =A C ' + >

( = ( ; * n = ç / + (=î - i ) , t : i ) * l ^ - h n t. : i ) t : - J ) = t - ; . - , I , i ) q )

*^- Àc =[: -l) t;;). (-f" ;*) * n --h C RtL 14+L "

,rtr,,V'

, , r ) [ n , n ) C =A t  c ) a')

un,h,t) --k A cl-,tn,g)=

r)

A.Dn: A

. A= C

C

$ tç,*^)

\

( I' h I

= ]*4"

q)

(-A+ A') .Yb= Afa+ À'{"5

q ,)

,A.tÊ*6t)= A'bofrg\

e)

[ o D b = a t À Ê ) =+ [ " 1 6 )

t^r[^.Q*}- l,^^*1,[o'*J,+.t f.-t ^J

n.t{"

d^ f"-I^

'

Pt--A' A , A'-- A^4"F ' .uÀy -l..$*i.^., {'"3*î

canry'u ,\ f c4^.

w^w

pucct'r'niw- J"l^* ^*l*tC

A"=T*, É^t- Àu A f*

(À^I-h s'N

f t L =A ^ 4 , - ' " A

A*l^t^^m^F "'t't- )

h- ?-I/t . s

A'. At, ûs^* ltr^)t r, s euV "

e H n A n ( q'-).

lrg&* i \,^"<,^,

*Jr

J*

l-r"*"Jt 4€-( "

A b {* ^J;.o Jq

^) tn' Tr Ab =

["ji)

n_.jsur ^-<

t YrJ,t"

(*).

e rl nvh

I .( hn

At

Ë{,(, €^}-th{y\ur-q^ dv^.*F w c$svnpaUk1,*-t V f ca-y0t^ cr\ .

ô. Ab= A , I,<,^ th^. ^*,1** *,^in c^J'Jilc W;t""f" V,',ie\a,^u\

-:

;.)'*q-

^b

b=

'=

È

T{n- elùnn + .

a)Y*^

CÀ

.^

16r

r v^f ^-.*.. 0. ' færavcn^''l'r-'

pf.)nar.c-tlut

A , { 5e F { * n ( k J o ' k q , f ' ' V b= o l R b{ ( o xA + f B ) ,-

&; 1 )

u\^A Èb

r

r0.lr.ll.

!

\

.? )

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Ygt/w

e (,, I v"-1 un o\" {à {.1. 'r' A e t l ^ ^ . ^ t , t ) i t'yr n f

-= n) Ca t-

3 )

P.^^^ A e

R

lt ç^6r,tlr 4

Al=.

r $ ^/ r *

H-ntù)

(A')

d^ ïfJl-

b,

o=

A

g' N = Ô'Ê':r" -?**-t"- A-o' J* i A 6 {';*t'ut .l^ * J*,y.

*t.^^JtJ ùs^ \--0''. Jt fî^ tut)

no t*Ï A= çi^ ^) ,,"+-{& t ï)" l{ffîï] t" {b= t J .,^kitÀJio^.

tdt îî^

{'cut\"'tù G Jttl^'e-^-}u

L

aI- (' h=r't ' O^.^"^U-

ft6=\

& . { \ =cr:*)t l î ( , : : ) ' - t : ; ) . \ : : )

( = ) [ ^ * "

\

,/oc

1 c

l "*gc=.

lz, o

. L ]

J "ti

%(

\t)

É) *-'fc À rtsrrt^?"'î^^"

c!,- c^ "^0^,'R- (: i), (a.'

-nd*lr, {'d\ ,-Qi b .A i* .

È h=t t

Û^ "^-

^ o ' n=) 1 1 = [ i)(: ;)

COr* c ^ . ^ - 0 ^A'- rn, = t n o J"-r.x-, E>.q.,rnrr f*{oU) . 4t ^

. - k ^ - R ^ *f^l = t : i )

i)

o =( ; F * L ( , r y *h È ' - q = B A J"^^-,

ca tsrÊ=1^=> (, ce,^ î:l

ft* t ru) -{"- v'^J-c-c

î)

( ? :).,f i ; )

=,('.îiii ô)-t \à Â .'ah tc/.,t;*0;1"4 '

c-c. d *il"\rd L'( , Jn^t 1.,,,r

l\,'^*ï"*, * Lu,.^JÀ.q

,-o'(trt Îv, *k u;*;1"4-

J.rr. \

fSù'=

Sh

[1.*1,,^* L"'!^-'an\ C. r . \ **-t-0. u *^,\ù L* t-l.Crl"\ "

h'.rrr,kçan

B ' Û F n t r - . , ) =* n " , * u + )'{sl'Ms . , 1 - i Wt *.Q^-

p^.r, t^*+te

-^-

rvr^^'Q\^^--e

( n u^w"^|,'&) , A{^.,- A-o^}'-

ù"-r; l"&

A qts ^ ^h.*, ù *e;l'"0"J*^â,u ( 4 0 - o = _B - nA '

4)

A "* b

I.,t{^^k

Jsr,. v ^or\À *, tl..* F, tq-\) &hC u/v\-e'

J'urfu vr ' M'w ^^rh^"* ù.'40.Â\"-{-

A C - - Ê C: )

A=À

b4. /tq^" Qu h,w"{.tâ A o^q vnJ;r" ù "^q^ry; A ^,1^w+o-'\,{q *rt-.n ^

5 ) !il.

IRt) l)*eorn"Q.

Ç**l-

(.+)

o

ç y-{,;rvr rr,

E ut^h n. fl,'Jot

- t=

t

' nn*hrri *,Q-L .I on* ïf-L "f ttnoaé

c^^ÀY

Ë ert- F -l.aa*

f

r1: -[.'a

. .J;u*

/q l^n ec,^'n

l

L

I

\- +rpc.rVzr.reJÂ^'"l-

J"

J*+^ç"".

u^ -,tln \ 'r^" ï"t'utuB ûe = 'ok r^t''*"l' ,,,*1. ,n n-l*,,bi"tl'^'-l- ,, \ 'F ç i { r e - . , F " ' f ç u n n , . ,k be= 9,,-,.*^ \ ,r^""ho"a<t'1. ' T

(un)'- , -t tun\ Êt^h J' v-"1û^ ( 0i o,*t"'.nr-r , "J'nn Lt eI' q f f4i. ^k^ç* a i=4r ,hl ù--4;'"rh * oij ..'0 eu\'& Jorr^ {: - A,^nYi/ ^f

v* + SnrJ*-t Rtun) = e,,r\ {tu*\=

1a{^* \n{{+-

;

-tQ* n{* - {

q*{-{*,

tan^V* = o^^ v. * "\.n{x-} r,) tu I ttu*\ ttuo'\{t-\)

0*

{;*Jù* "-trr^ "Uh+J-

shir*"lr

d"^

o'I,[Jl",

* *,,bt^-,rctrroc"-ei. À (

CI^ .{^*Jc to'^-

t(b_ *,1ûr It

qt^f

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1

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ng I Au, t*). {^n'tt* r-*-

fÏ'c'J,'o^

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g,Kt (.,ï\ -D

{ à ( t ,1 ) =

-dh*tb)

(**1r"r-$r

qr'r'l c ha"

.ïrJ^ t^ { ^IqS:,^Îâi dï,r'H':.ï'f,,

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{" -1 d^^J,"+ îo^s^^t*

c,i _ ft+l

Or'.to

:. = ,(*u,

- L ( + o ) =l -ù o n + r * v o ^ { \ } I

, e/n 'l*J*

Jn *d^{

d'f 'A r'L1^r :

1)-- dtd'r+Éç+Tur (tu*) = ( 4'-4'

Ë

t!+? u'' [ ( * \ = . - ' 1 5u n - i o

&" c\ ."0^. t

(t,J

\.r

t(c

r l4

ul-

T\Q

q)

t^b

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tlvrc

Le ''

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c,.*,lrFl4r'edrun , . J

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il'

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X Jsviy*

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t ro)

À L{^..a*1É,^lç-

Qn'durE

n R3 g.'. Kq ( v , u ; , ) , b ) r ) g ( n ' $ , t r , 1 ): A A

A ^ =( i ;- t \

e,

( t ^ . î ï , 1 ) . - ( ' ' n - 1 + ? b lrt ' t + u ï- ' , * ^ - ' l * 1 7 - A b )

(( oJa â-r,wr1,5nJ d^À

puu.4'' Yf-0i""fiù'c Q", 1'[o^r"La*^

0n{.'i/+o

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t-\^

B"t

Nr,

À6

È. = H, (Be,e*) + ir( b*,s*) t- Ê. t 7,' ' ) €

I B = , 6 F ) = .À " s ç ( 8 E , 9 * )

i

l

K - e1r*.r t. oLûu'J Jt bu,,Êa * {àc "{* bc-,u

€ h) t ed(G,Ê ?*dtF,c).

( Êe'{36) = ù\ ( û e , o u ) (tr"B) z il ..-L,..*

o\t t-

( û,,6J.

\nr,vnuq-Suiytttl .

&*A^< . 1..1'^,JrÉ .^ F .L^u \- Tc^u v"&niJ

t#-t*

gÉ 'ôF Ja^ V"'xn t\- V .kF z'-1"e'A.,t'*",[' ç'd{^^l*l..'*a1.vù ^ . p--hrx*A*k '^)' l;y'A* (r"ruto^n' gç4t6'tr) W*ïÉ.Jno^ D " YQ^' t(( [6c '6r) *'k;w;\'-Q" ' /\ ^- 4

= ng.^tô*,cà*) [ngt6e,ur)

(\

ï\.r,trg".,* rt

pJ{h* ï\i,ri-

ffiJ,,,u

€',cls- "'\' ('u* 1r^r*

s'^

I

hcv< À

h*u'o' "4 r

ù

'lo "L'p"r^"* ^), 4 , uu J^n^-J e1'c.tr, KË u/^

sLB'-È =Tu;\,=l l d;\rl,. c^YrJft v"n*ç-

,n *Ï* * ,L V^ o t

{o * *lÀ^'{' t''n'n{'.'

-{., 8* VW Y>À Q.- 1..^Xn) J"^ e') A*t {' u;'l'Jornvr*â ' Q ,n ^ \ t o n v < T u , \ , t1\ 0

l,tù

ulr

h?,^

r> t' (

Pr-g')

\ t=

6

A

-[0r." \'*

çadr,^b"

qrt Â,r), .{*\4,.\ \ i l t *ç n , n ) \ \ ut^\a,.),

Tf".=*"t..- 8 , * - o g , )

(fun ^

R ",

Jtt^

C*.ô .tii-Lrt^ (À-

L{"

u\n

lt'. a,6 e ,f' 1 d . - +a h \3=

\o= <at4fr

t44 )

(^ ,_

t'l = o((,{Ao f

W J"^.i"*

d t & É

^ = +d,+lfr

''t,- 7 i , c h . 1

d = ' 6 *,1,-

:>

fr-\

h'1

p< vr^ ê^r, û^ J^.c^'D'""(r\.ef- 4 :

5 _j

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d*

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\o,.^*I^^* ùoq"-{ -*

J "*

( p

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t l.n-, ay/

? ( ff\-, o)

s^- h* d"*rr q-i,st Jtn '6L e E) tU -' t^\* {n-'-' eâ r{t\n .'*^ J us\N J* tk '

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cr\*

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h

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d"^ c. *Ar-

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fit-

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J'on J.*^y,*J-

E {{.;F

ïf{t.r'-l^\*' "*,.*-r-.faELk tr

JeF

-t

g^^ -q^ (.,t/t{

P .X X . = 14'-r f\

Je Lo,q€ Æwr

v,^.'}c-q J'u"^-t

0l*u'.* Jq " *L *l.t^* W-e-/Y<c.ç..f..,h,t

À , i , L( * d { . ' e , p ) ) V.*

{.".r.,,Jn.*

É tt

-[t^',-Un"vt,

B e = \ u n - , u " \ " t s Ê t u = \ u ' n - ' h t n \' l * ' * x .-9'u*l"olnor et=1v\o-'v\\ 6F-\un-''lh\*;.

&" *Je'

A = ilg (6='t*) â '= ilt tstu , 6tr)

P : ?(Ê*- gb) q--

P,

o o- , f i * t )

(9' c^ .^Ln^

L*v*

At= Ë=-

q-'È P F ) {*dte)

ôç=\qn-,u.\+

Atu =ldo-,tilr\,

A = Hg ( o*, 6Ë) t

a\, ?=

At= P-n À'{'

r*,*-fo _4"

A'\(4:'d.)

-D

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