Marius Burtea Georgeta BurteaAna Mihaela Alexa, Rodica AlT amescu, Aurelia Bdlan, Cdtdlin Birzescu, Daniela Calistrate,Radu Cristian, Teodora Comga, Gheorghe Dicu, Costel Dumitrescu, Middlina Enache,Elena Fifu, Cristina Lemnaru, Doina Lorant, Sorina Lupu, Simona Mdgureanu, Sandu Nica,Mdddlina Nicodim, Gheorghe Popa, Elena Popescu, Vasile Popescu, Elena $erb

Teste

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Cl tSr'\ a Xl-a

MATEMATICAProhleme si exercitii))

semestrul Imatricedeterminantilimite de funcfiifunc{ii continue

servicii, resurse, tehnic

CAJ\€PIffiTq

o r' (s r-)=B 'l l=Y

- rr \€ b)

T

elrJlEhtr

e;in1o5

- .ttq * ,zzv = x :{ePc1e3 (P

,urele eums ozelncl€c es ES (c

d ap alalueuele olJcs es PS (q.rereg pdrl azeztcerd es pg (e

:3lecr4?ru nBp os 'I

rP z€oluJoJ (tilD '"''L-u'rD 'ULD)

€urnu ep lBuopJo Inuelsls .rou as (3) "'* lr{ , y pxee .

)=V e ):8'8 =Y o

'F-r 'Y=8€8=V o

l:\"-w)YA'Y=Y o

ld ele rolaculeruuelelge8g .l:l'"W = g'y e1ect4ep1 .? f-eef,I4€tu u-w qxe(l .

'DU0A9) a)ulow

n (i'u) pdq ep rccIl€IAI .r-

D Z,uD I*p)

I

. l=V .:D z4D ,rp I

I

.-D ztD ",)

x ,u'tu) lndp ep r""ro:il"'":arunu as J)

liD eleJolunN .

: 'li : P' eflcmu o '3'eeutflnru

ndp ap acIJlEIrr a$eumu eg .Jparo4,tDliatg

'IITY9f,'IJTUIYI^I 'I

16"""""""" """'ogur8orlqrgI8""""""""" "'runsNndsvu rS lllv3roNr

08""""""""" l**r*-*91""""""""" liectldy'[B^rolul tm ed rricunf Ieun Inuuos '€

It""""""""" 'onulluoi lticurg nc 1$ered6 'Z

S9""""""""" lcund un-4u1 enuquoc rrfcung .l59 InNIINOJ IIif,NOd '111np1;do3e9""""""""" "ilffiprsEp6rsql65""""""""" ...'...olue.l ropdcutq rn1ncgu.6 olelofursy .g

SS""""""""" """'lllcury op rolelurll lnlncpc uI apldecxe unze3 'g

8t""""""""" '$cury ep elIIuII nc l1|eredO 'n9t""""""""" """"""""""'oretuetuala roltticury elelrurr"I '[9t""""""""" "'..'......'...o1€ro1ul elrur.I .1cund un-r1ui 11fcwg reun 4unl1

.Z

VE """"lQtgutce,r 'etutSlptu'a1uue1ut :g1eer eldearp ad epund ep nurilntu erdsap oreluaurale runrfop .l?9"""""""' """""""[I]No.{ u<I sruluf -1p1o1;da3

t9""""""" pcpBrueluur EzllBuB ep aluawalg9t""""""""" "eJetllelo ep elsel06"""""""""

...-..... anpuq8uruty atato"rdns taun DLIV .e .Z

9e""""""""" ......... .aw&tD!u11o1 .a1cund pnop uud oqda.tp o{oncg '1.7

92""""""""" " ouleuoeE ug .ropfueuruuepp ele rrfucrldy .Z

61""""""""" "' tfplatrdo.t4 '€ 1[nru Ieo Inurpro ep ecrtrarlgd ecu]Brrr roun Iqueurrrrr.r]eo 't6I I[NyNI61UAJSO 'II p1o11du3II """""".'" "oranlele ep oisel0l""""""""'" eculewncrriersdg'gI """""""""' 'roleclrluru eep111e8g'ecr4ehl 'l

SCtUIy14t .1p1o1.rdu3

YUSflC'rY

SNITUdOf,

1. MATRICE. EGALITATEA MATRICELOR.

Breviar teoretic

. Se numeqte matrice de tipul (m,n) saumatrice cu ,n linii gi r coloane cu elemente din

mullimeaC,ofunc{ie A:{1,2....,m}"{t,2,...,n}+C,datdderela{ia,n(i,i)=au.r Numerel e ai e C se numesc elementele matricei.

o Matricea de tipul (m,n) se reprezinti sub forma unui tabel dreptunghiular cu ru linii gi z

coloane.( u,, atz ',, )

, I o., ot: '.,1A=l -lll[o-r oo,2 a,,, )

r Matricea de tipul (m, 1) se nume$te matrice linie, iarmatricea de tipul (t, n)se numegte

matrice coloand.o Daci m = n matricea I este matrice pdtraticl de ordinul r .

o Matricele A,Be M,,(C) suntegaledaei qii=b,i.vi e{1,2,...,m\, j.{1,2,...,n}.o Egalitatea matricelor are proprietd{ile:

o A= A,YA€ M..,(C) -reflexivitatea

o A= B > B = A, YA,B eM,.,,(C) --simetria

o A= B, B = C ) A= C, YA, B,C e M,n,u(C)- tranzitivitatea

o Dacr AeM^,,(C) se noteazdcu Tr(l) =(ttl+Qr+...+ailil urmamatricei l.o Sistemul ordonat de numere (arp err,..., a,,,,) forrrreazi diagonald principald, iar sistemul

(.a,,; a2.,,-1,..., o,,,) formeazi diagonala secundard amaf:icei A.

FNf S" tt" pi _r :"7-t_!g!:

t9....."......19

81

97

1. Se dau mafficele:

(+ 3) (t -r -2)A=l l. B=l l.(-r s) \3 0 -t) '=[:JD=(-'1132)a) Si se precizeze tipul fiecdrei matrice.b) S[ se scrie elementele de pe linia a doua a matricelor A, B Si C .

c) Sd se calculeze suma elementelor primei coloane in fiecare matrice,

d) Calcula{i: x= arr' +brr2 + cr,' +d,r+dro.

Solufie

Matrice

rz-: =a:sru1l:]l*1t1..... 11

.....................43..........................48

,,'.....,'':.,......::............:"..:365A<

.....--80

alrJlEN9or"ip ul ep riqllm ed rrfrcrexe rS erualqo:6 .€-IX u VSVTJ - VOIJyI TAJVI I

:-.:-QEru 1ndr1 ezeztcard es 95 ft(o I r)l, s tl=s tt\i z t)

;ectr€Lu ap ealeltle8e col eru erec nrlued eluoJ olaJeurnu eu[rrJepp es pS .S

., =, ,rro.(t ,l= n(r t )

(t e)'l ^

-_l=V pceurfqoes .t=z( erinlosnc {_91 ={+, erfenceeurfqoes zzq-z.D\€ t)

ualelrle8e UKI 'l- = x erfnlos nc '9 + rg - x +Z urfence s\nzeJ ttq =

ttD eeplrleEe urq

a$n1og

.e1u'e *ms (n -y' n'

)=, \t-ltZ 9+xS)({ +t r-s) 'i - l=w elecrJlBr.uerecn-rtrued uetr,r rol4eure;edalrJolB^euruuelopesps .nI E ,+z) '

'9-=c IS 0=q ,t_=o.g=rsprfnlos

no'r-r-s- rs 0=QZ ,0- o-.E ,r=x-tapfence"lrc"rs .[j ?)=,

Bocr.uer tr (q

'S-=c IS 0=q ,t-=D ,b=xg3PlInzoU'0=e-"9- IS 0=QZ'0=D+e '0=x-b pc?pglnuecr4erua$e v eorlJlel/{ (e

a1inp5

L'I=v erucn-4uod x')'q'D uolelglsrxg (q'elnu eclr1eru else tr erec n4ued )c ,J ,q,D rolr4sure;ed oluol€ aur.uJepp as pS (e

("-s- sz \ '(u)''rry = | l= 7 eeil4eu erg 't\.'+t x-n)

!- = - Pluer erfnlos nc '6 = 1+17I^

erfencseuriqoeg't=€+€+l= =rrq+rrq+trq=(S)rf ,L=l+t*E=tto+zzo+rro=(f)rf

:lrms Jolecr4eru olor.uJn (q .0 =

trq ,t = ,rq ,l = trg ,rrlcodse-r

'l = ,,D 't = ,,o 't = ,,o luns eJepuncos reluuo8ep a1e rS eludrcuud rsleuoEerp elaluo1llelg (u

a;fnlog'rlk7rt-rr)rr]= ,"Q+ r"a+r("q+t'r) qcpur4s 'u3r rnlBoreolelerrurepposgs (q

'8r IooI4SIUu pJepuncos uleuo8erp sd ep rS y lclctwuJ e pledrcuud uleuo8erp ed ep olelueurole erJcs es pS (e

(t r o) (r o o)ttttJ0 i €l=8'l o t ol=tr:elecl-tluurEreprsuorss 'z(r z t) [r r- t)

'lt=Z+€+20+r(t-)*,s=, (p't-='? IS 9=6+t+g- r€r+ t1,x+ttc

,V=t+l= rrq+,,q ,t=(f -)*l =tzD+ ilz eleums urfqo eg (c.2 eacr4eur

rulued'1ig eoctqerun4uedol-'0,giV eacl4erun4uod.S .l-:ololuetueleurfqoeg (q'(u;'''^3o'(u)"w = )'(u)''tztrrs'(u)'''w = v pc ruo^v (e

= f- :elocIJ]€IU EJeplsuoo oS '9

J ?aciJl€Lu eulwJalep es 9s 's

'ql = g BecI4eIu olJcs 3s gS 'ttrl = V eocl$€lu olJcs os PS '€

r g'; ) lndrt ep qlnu Boculeu (;:eusoloc I nc olutl eclguru o (t

:elJcs os 9s 'z

. n, (t'z) pdq ep ect4uru o (P

e nc (E't) pdq ep ecr4eru o (c

a no (7'6) FdIl ep ecr4eur o (q

> nr (Z'Z) pdrl ep ecr4etu o (e

ts ts

:oucs es PS 'IF=.

)gc ureqiqo elmlcuoc uI

g=(1+,f)zEol etfencg.E

= l/-a1s:

Ltrr=9-u-(u nBS;f=,

- (l - t' l:''-. t:-. l') rr 'i ! / nes [= 1;; :arserrrt

'. '. = ttt nes [- tu( = tu PJ ]ull- .

, IS rl = Z - x + zr BruJoJ qns tl jl'!

= ,r.- _-\: :apfpyle8a qfqo e5

a{n1o5

: -((r+'{)z8of il: ,,2 r'* ,.\

f'--

.Tr:- :trtncea B ; 1, penflu:- r

---- -i.

_:_._11

-{ s: re pe diagonala secundarl a

- .,,' = [rr(l)-r'(B)]'.

> jl-r: J - 3, a.z = 3, a., = l,

1 - -: - 3 = 7. Se ob(ine ecua{ia

esle maffice nula.

t-r 5r -5 -c=0.Rezultdci

lr=0 qi -5-c=1,cu

ame eie

i malnce:

Ce inr6lare

' .:]succeslv: L, =J

r(ir-l'):=J. sau2

este ,? = 3.

Exdt#A@Jiiil' ,:

1. Si se scrie:

nl.S&U--=J(n "2)! 2l

(r'** ) fz ):a-r)I l=' I

[.; rog.(y+rl,J-[: 3 |Solufie

Seob{inegalitSlile: x'+x=2; 2" =2"''; Cl, =3 $i log:(y*t)=3. Ecualia x'+x=2

scriesubforma x2 +x_ 2=0 $iaresoluJiile rr =1 pi xz=-2. Dinegalitatea 2'n =22tn-t ""

ob{ine cd m = 2nt-1 sau m = 1. Folosind formulele combindrilor C', = -A* se obline

(n - p)t pl

(n - 2)t(n * r) n

(n-z\tt.z = 3 . DupS simplificiri rezultd ecuatia

n2-n-6=0 cusolutiile n =3 $i n=-2. Deoarece n eN*, solutiaciutatl

Ecua{ia 1og, (y+l) = 3 se scrie sub forma y t | =2r qi are solulia -y = 8-l = 7.

inconcluzieoblinem cd x=1, m=1, n=3, ./=7 sau x=-2, m=1, n=3, /=7.

";j p1."!-!e.

-m.e pY"P *!:-

.: ::::::l=:r:' ::::r,:':,..'...:a::a:. ::

1,.:.::::::::.:,l l'::l=i r,l r':'ffi,5i Exercilii

(2+x 3 \.t -l I-1 - | t.(5-x 4+y)

Fa r=-1. Din egalitatea

brineca ,=(:1),

a) o nratrice de tipul (Z,Z) cu elemente din mullimea numerelor naturale;

b) o mahice de tipul (:,2) cu elemente din mullimea numerelor intregi;

c) o matrice de tipul (t,:) cu elemente din multimea nttmerelor reale;

d) o matrice de tipul (Z,t) cu elemente din mullimea numerelor reale.

2. Sd se scrie;a) o mah'ice linie cu 3 coloane; b) o matrice coloand cu 2 linii;c) rnatricea nuld de tlpul (2,3); d) matricea unitate de ordinul 3.

3. Si se scriematricea A =(or)eI1,,,r(R) gtiindcd: a,, =(_11'*i, i e{1,2,3}, i e{t,Z\.

J. Sd se scriematricea B ={b ).Mr,r(Z) stlindcir: b, =(i* j)' ,ie{t,Z\, i e{1,2,3\.

li+i.dacdi<i5. Sdsedeterminematricea C=(r,,)e,lZ,(R) qtiindcd:., =]l + j,dacd i= j.

I z-i, aaca I > i

18 3) " ( :7] .=r, I rt ,=f f)6. Seconsidcrimatricere:o=1, u).u=l']l''-tz 4 3) tJ:.,/\ / [8 0)(1 2 3)tt

E=14 s 6l

[z8eJa) Sd se precizeze tipul matricelor A,B,C,D,E;

Matrice

alrJlPhtroreip^ul ep lipprn ed ufrcrexo rS eunlqo.r4 ?-fX € ySV.I) - yJIIVTIAIVW

lal{I3luela ewllrJolap es Ps 'rz

r-,^ ) (i(r-r) E ) l=l . I

t-x).2) [ r i(t+r)Jau oletruoruelo deuuuelaq '97

'6 > z',{'r

:lI

t l= v alerl-sEru rcp es 'sz

tll

( -a+ ,u-g1 ^tutz- '.1

r M^Z tl+ zi+- z^ )

elecl4Bru eullllJolap es ps 'rz

(o z+x ) ,r- - l=y (e

[r (,r+x+1)qJ

p gcr4erulsuue elso y PcrP4gd

:l?ar In.rgumu 1iequ.ntoq 'ET,

l= p aprm "O= N qJE61 '27.

' x)l= w ve3. Ji,Bur ?p os 'lz

,Z)

( I (l+r7)s1)"/=l I (?

' \'' ,' z*'1te )

F\ (g+* l-z Z+r) ,. .=l - I \-l, [ri+l ,t ,x )

I)_f t z+zi8ot) {E

7-,;- [s- ,Z Lt- b )

,i?er eloroIunu dumuuopq '02

{p U ",('don liequlrePg'61

: g = (lr)"rt erec n4ued r ml alr.role^ $eurruraloq (u

( o !::l (r-{)rsor)

i ,u ; , l= H '(u)'r'v = tr Berr.q,ur arJ .8r

[tt-,rl;ao1 t- .e )i11{fi112.-- i rXiiiiiiiilll lilXlii.it r.iiiiiilrrffil,f;;firi}i=;7fi-+:1,, ,:]iip++?l 11;,Xry r,p

(e** r )[ , n*o)=r

* [z*'ie 1l=], epun.eluaeluns s $ vetacrreurqcsp u >t,x ageasEs .tt ' l. g t-xz)(,1- ^)_f r r-c\[,*, ,*,J=[,*, ," )"'n"1'Eeurp

33/t''? aulrrlJe]opes?s '9I

o) (e o )s l=l t-r 8 |

eelaqs8aurp g ) )'q'o ourruratepos?S .SI

*z) l' t z+Dt)

,t ) ( o r-r)irz)=[r-, r+D)BeNrvEaup 6 ) c'qo, oullrrro]epos Ps'tI

e+'l_1r-r i )

lf ) ( r z+x)v,4leSo urp g = {'x euruuo,.p es PS 'tl

(: r-) (t ?-)

t; , J= |.r, 1 )n"rorn'. urp O > 'r'x autlure]ep es PS 'zI

(o z-\_(o ,{)[,- ,tJ= [; E- i "l't'tnae

ulp z: '{'x outure}ep es PS 'tI

[: :)=C ]) ,,,n nnr" urp N a *r surur-ro1ep es ?s '0r

(",

ler[,

(z- o

[,.(rIr

rse : rnl aluoleA [ernuuo16

e i m1 al€ IJoluA ac nluad

.e -r rnl alE uols^ oc n4ued

'?- nus 7 nc epSs alelua4lele ne erec (t'S) 1ndu op rolocr.B?ru Irupumu rleunurspq .6

'1 nes 6'1- nc eluEe eleluaruele na eruc (e'z) Fort ep rolecr4?ru lnrpumu $eqrurepq .g'q nes D erelueurale nB ersc (r'E) pup ep elecr.B,ru elnol er,,"s es Es

.L

'(r)a+(r')a empct€cesps (q

: (r)a nc plelouts recr4?{rr uurm) 7 rocr.nBru e gledrcuud eleuoEerp ed ap rolslusurep

"ums ozelnclec es gs 6

lar pctIuru s pJspunces eleuoEelp elJcs es pS G:cr rccrtBru rolelueluele aums erurcez o ep pclur IBru eJsoJa nc gsd11 uud szetuxordu as gg (e

i3 rscl.Buturoleluorrcle sums ezapclec os gS (pig eecr4uur qp enop e urml ed ep eplueurele erx)s es pS (c

!y recr.qeru u gledrcuud eleuo8erp elrcs es pS (q

rrna elementelor matricei D;

mat-icei I (urma matricei I

sai:.te-e esale cu -1,0 sau 1.

:ele egale cu 4 sau -4.

)2)

lr-l 6 \= lsi3 3t +2)

;ttl:.,::I I :a::::::.,:,i:i-:

,e !nvdtare

b) Pentru ce valori ale lui r avem cl, + (t.r < a21 2

c) Pentrucevalorialelui yavem 4r3 -arr-arr=0 ?

d) Dterminaii valorile lui e astfel incit ar, = arr.

19. Determin ali a.p,ye IR din egalitatea matricelor f sin30'

{^"t ( " ll

rLtrtur L 7 cos3o" )= [rrro Ji]

Matrice

20. f)etenninali numerele reale x,y,z gi r dacd :

( c:-n r"-s') (-2 3 ) (--:a)

l,l*,,-'', -

, -J=[ ; r*,)t b)

[l;l. (*, .y, r+r,l_fr o ol, d) (*]-r,c)

[.rnz z-t x +6) \4 2 s) \" -Y'/ -tr-z -2 - \

e) I " '--'l=1,.' [lg(2-v+l) | )

21. Seddmahicea * =l'- 0'.}. anuti ,r.1'elR. astfelincdt M =lt-

[x 4')

22. Dac6 N=ororrnde t*i=ft'..!l*') lo*'('-t)l ,sdsedeterminenumerelereale r qi y'

I o log,l )23. Determinali num[rul real x astfel inc6t urm[toarele matrice si fie antisimetrice: (marticea

pdtratic[ I este antisimetrici dac[ A = -' A)

a) A=(tn(t+x+x'1 ''1, o,r=lttl' -;:, :l( r+2 o) [_z 3 o)

24' s[ se determine matricele * =(,'* l) t '''o'

qtiind ci are loc egalitatea

i'r-w:+[3 2vw )-(+, ar')

-2wv 13 - u'2 * u' )- \-+w a, )'

(41-- -2'l ( z -!'l l25. Sedaumatricele e=l I rF i si a=ll-:l + l.gtiina cit A=B detenninali

[ ; "2 ) [* ^1,,)-. .. - -]D.i. -| ri E !\.

26. Determinali elementele necunoscute din egalitatea matricelor

1 r+ r)l 4 ) (2.1* -r1r 4)

3 (y-r)r.j=[ .y'-r t)27. Se se detennine elementele necunoscute din egalitatea de mahice

-.i .i

_, _1C

a)I.IlEhlOIoreiP ul ep fpllun ed tdrc"raxe rS auralqo"r6 .e-IX B VSV.IJ - yCIJVIAIAJyI I

. rrp,l= Z+ l- = rtq-zto - ztpt

P ''p)p ',pl=O=g-y pceq (q

? "p)

1 * (g +r) ozolncl€c es PS '7,

:rp .l = (r-)- o - ttq- tEo -

ttp

(r- r t) 'l l=-)Ir t n)

= ::c iy= 0+t = t'q+tzP = Ii)

t =.\c,.y= I+€ = ttrqltD - tlc

J Eecr4€IAI')=g+v erl (.e

e$n1og

,'r /o t t) -s'l " -'l=v (r1 \I_ Z E)

-r tS g+/ ezalnrlerosES 'l

'(,(efe3-uoqtuu11

- =F,'(J)'7,t1=ygtuq .

' ,-,uV = ,Y' *V o I

':, '11 A 'ellrfeler rol nv .>uliep as .N " '/ n4ued .i '"

I = oP' etseuueP as .'(a)'Yr)Y ell o

v,'$, =(s'r) ,

8,*V,=@+f),:e1ule1er col nV

: i:url uj Joleusoloc e tS euuolo'^ :r,- 1" (A)''*w)y gxB,C .

' i.,' = t'' A'V = V.'I =' I'Y

? (eletpuapl) apqrm eecl4€W .

'(a)''' lrl = r'(J) o'"

w.g.,(o) Lt'trt

vq1 ) y A,) (s.r) = (c. s). r:pl4ercose else earrflnurul .

: rut!1nwq aliplaudo"t4' {d'"''Z'fi > l' {ur'"',2,1} ) !, r"q.'!D +... + t.q. zl, + trq. trD

=,t!),rc '(O)''*WrJ,(,b)=J eecr4el4l (,5)= S ,(,,o)=r ,(O)o'*W=S .(O) ,,*14{=y eIg, .

rop)t,qDw oat{lnutuJ'r? InrBr.rmu nc ,/ lacr{sru psnpord elSeumu as (r, n) _ V peeilJle]yl". (J),,.,,,ru ) v ,JD l? erg

Joutnu un n) aculnu,t oun oat{putu1., *o

= v +(r_) = (v_) +v;uelelerrdord ere rs 7 recrrsru esndo elseunu as (rrr-) = v- Eext$ew .

'(a)"'*w>y A'y +''''o= ""uo+y:aJeunpe nrlued nr1nou lueureJo else ,'.p ulnu eecr4BI I .

'())""'w ) )'g'v A'(c+ t)+v = 2+(a +r):p^rlsrcoss olse J0lecr4eru seJpunpy .

:tuDunpD alip@rtdot4'g+V pzeelouas 6r rS y rolecqgrueruns .

'g lS y roloor't1etu uuns elSaumues (ar+ nr)=r'())"'"'lnt>^J €ocr4uru ,(,q)=s,(nr)=r,(O) o",nt4{)g,v gce1t .

?qaloa' totiatgflJIUIYI^I Of, IITYUf,dO 'Z

(::r r IB = h' vrnr,,rS | ,r- 4 r l= s

t-; ,T s-.v)

'z rS r + 1 Jolorerunu e pculeruoe8 elparu (q:lzrl 1s lrl'lcl'lol ;olereunu B pcrtrourrre erperu (e

(z _z/*r) (t'3o1, r+rA).tttl :tleururataC ' j 3 gtz l=l ,Z+ ,v ,_& | n.rnuln8, proprsuoc eS .62

[s z ) \,qe-t DZ-t)

'p1 > a'g'rzo,{'r rfeunurelep

( x rrA- -Fror') r-l tS la+t q-o l=Z

elcrrrutunepes .gZ

\q+o 8- )

.(=z lo*,r1 (.oero, ir )lr sor ,,J ['lv*1: ,2.,.,t)

A'1, = 1,,'A= A, V Ae M,,(C.).

r DacI AeM,,,n(A), A=tor), matricea'1, oblinutidinmatricea A prinschimbarealiniilor

in coloane gi a coloanelor in linii se nume$te transpusa matricei I .

Au loc relaliile:$tiindcd l=B

);:e:::lnati:

(c), c =(au+br) se

, '(.t+B)='A+'B

' ' (.e'B)='B''t. Fie Ae M,dC.).

o Se defineqte Ao = 1,, At = A, A2 = A'A.

c Pentru r e N' se definegte matricea An : An't

r Au loc rela{iile, Y m, n, P e N :

a . A^.A"=A''*", '

r Daci AeM,(.o), o=(' 1.l,*. locretalia: A2

[c d)Hamilton-Cayley).

'(.q-n)='A-'B

'(' A)= t

.1, numitiputerea n amatriceiA.

(n^)' = A^, =(Ao)' .

- (a + d) A+ (ad - bc) I, = Or; (relaJia

. l- r'

.

si A-B

(t -l_t-[o -2

(0,, ',,)Dacd A_p=p=ld, d..l

[r, ;,:,)

= Qrz-brz = -l+2 =l, d", = al

1. S[secalculeze A+B

(t 2 -l\rr A=l l. B[4 3 0)

Solufie

r\rl-2)

"'' ), *d"c';.)

l;

1)= -t' A;adar

dac5:

2\t.

-r I

(2 -1) (ob, '=[; l] '=[:,

cn = aB+

l; cr. = an

a) Fie I + B = C. Makicea C este de tipul (2, 3).

C,=Qrr+4r =3+l =4; c,z=arr*b,r=2+(-t) =t;

L:t = a2t*bz, =4+0 = 4; czz = arr+br, = 3+(-Z) =

- (+ I l)t =l L[4 t -t)

dmatriceil cu numdrul a.

a C:(c*),C eM.fC,), cu

r_).

neutn.l

Rezultd cd c =('" crz

\czr czz

br, = -11) =+4, = 0+(-

= 0-l = -1,(2 I )

,=1, -r l.[r 3 )(t 2\B=l l.(4 3)

r)

d,,

se ob{ine cd dr, = arr - bt, = 2 - 0 = 2,

*bzt =3-0 = 3, drz = arr - bn

tl., = ar,-hr, =0-(-l) =1, dr, = Qn*btz: l-(-2) = 3. Se obline

2. Sd se calculeze (,t+ B)+t: qi l+"(B'rC) pentru(z l\A=l l.(2 0)'

ie invdtare 10 Matrice LI

rezolvate ffi