Mayius Burtea Georgeta BurteaGeta Bercaru, Cristina Bocan, Daniela Dincd, Latta Durnitru, Ionu{ Georgescu,

Titi Hanghiuc, Simona Ionescu, Roxana Kifor, Paula Nica, Ilie Pertea,

Mihai Popeang6, Daniela Podumneac6, Diana Radu, Carmen Rusu, Dorin Rusu,

Loredana Taga

Au$lisrul qgstsr a fost aprobat prin OMEN nr. 3022108.01.2018

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CL.ASA a Xl-a

MATEMATICAProblemeTeste

a r OO

exercrtrr)

sisteme de ecualii liniarefuncfii derivabilestudiul func1iilor cu ajutorulderivatelor

profilul tehnic

CAMPIONBucureqti 2018

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ffi:Ii?:i;:" {.-ie

MATRICE INVERSABILE iN

27

Breviar teoretic

r Fie A e M,,(c). trlatrice a A senumeEte matrice inversabill dacS existi B e M,,('iJ), astfel

ircdt A'B = B'A= 1,.

o Matricea B se numeqte inversa matricei I gi se noteazd B = A't '

o o matrice A e M *(c) este inversabild dac[ 9i nurnai dac5 det(l)+ 0.

Calculul matricei inverser Are loc egalitatea:

A ' = -)--. l'. unde ,4. este matricea adjunctd a matricei I .

det (l)Elementeie matricei adjuncte ,4- sunt complemenlii algebrici ai elementelor matricei

transpuse ,l . complementul algebric d,, al elementului a, este dat de relalia 5,, = (_1)'t' A,,.

unde A, este minorul elementului dr.

. DacI A, B e M,, (C) sunt inversabile au loc relaliile:

a)

a

(o')-' = o, b) (aa)'=s-t.trrDacd A este inversabil5, atunci:

o solu{ia ecua{iei AX = B este X = A-1 'B;

o solulia ecualiei XA= B este X = B'A t.6060636672

75

l. sa se derenninc daci ,u,ri..r. I = [] :)

, = [], :')

sunt inversabile.Solu{ie

o matrice este inversabild dacd are detenninantul diferit de zero.

ob{inem: det(z) = 5 - 6 = -t *0, det (B) : 3 - 3 = 0, det(c) = 0, det(D) = 0' Rezulta cd

matricea I este inversabild, iar matriceie B, c, D nu sunt inversabile.

z. Si se determi ne nt e* pentru care marricea , =(' il .r,. irversabild.[r, 8,

Solufie

Avem cd det(,a) =16-m2. Din condi{ia det(z)+ 0 se obtrine c6 nt2 -16 * 0' deci

m+*4 saum€A\{-4,4}.

Determinanli

" ^.ffib,ffiffi

(tI

c =12

[:

1 l\ (tr 11, o=lztt) [:

0 1)

I rlI12)

riuEuILuJalaOoreig^ul op Iiplpm ed uircraxe rS euelqor6 .€-IX B vSvTC - yCIJyI IAJyI I

soclJ}€ru 3l!I '8

r/ ole3ul€(u PJeplsuoc os ',

gr rnl €sro^ul deuriu-repg (q

:fg \3 SV [e1nc1e3 (e

)0)I

, 0 l= r' : elecrJ]Btu olJ '9

0 r).IE SSJOAIII

I)I

I i= F' eeruBru Pc du1ErY '9

t)(t I +'rrutuadgllqesrenurerse I r [-n I

lz z+x

0 r)t o

l=r' . z)

\ gY)' rgJI lUnS 3lecl.

; I)| 7l=V

-t 0 L)

'Y-'I=,(Y*'t) $ (y+'I)=,(v-'t) gcgtltrzoterfulerptsesceulc .21 =(r*rt) (r-rt)

'1e;trse'rS (r +' t). {v - z t) = rr -, I wq

.

rv -, I = ro -, I =, I :Arsecrns rueav

a;fn1og.olrqesrelurluns

U+rI l* y_rIeloJrrteu gc etem es ps 'b = zv gc eelulerrdord n, (x) ,w. v eecr4errr greprsuos as .s

't*'z\n(!'--l=,,\r )

erfnlos nc 0 < Z+wS_ eMZ , w u171nper8 ep erienceur Etlnzea .0, (g_ *Z){t_w)V_V=V

pc aurfqo as '11= xA,O* (r)lrp ulq .E_ utz+xZ_ ,x(t_t)= (z)r"p euriqo eg

'o * (y)rcp pcep plrqesrolur olse y e;,curelN

a;fn1og"!r

= r ectJo

'rY *evlrv)4eurpc fe1p.rV

(q

(e

:aleculelu el.{ ',

€+rr),*, l=,t)

=V(z o o'\ _ (z o o)

lz- t ,- li =,r,nt'lz- t ,- I

[o I- t) ('o r- t)'r=l' 'l=,,,

.u=lo Il -. lo Il

tr rl p rl- = "e 'o = lo ,l=

''eI rlc -t

I

lr rl

lr rlI l= "olorl '

.,-=[ li-=,r,r=li l='r.,-=|j l-=,,

eecrJlaru e;ec n-qued ,V,> w euruuelop es BS .?

arJcs os plcunlpu uecrJl€htr

:Joloculsru olesJo^ur azelnclm es ps

' 'lto {t 9l ,,.t+D t)' \t D)

ruolap agsg8 epole,r rulued tS

mlruleruered ellrol€^ {eUV 'g

)t ( nsoc purs-)t u '[ru,. ,*r.J t't.

3 (q {l :)[s aJLII€ru e4urP erec rfugY

[: i)=r '(: :)=,epcr:1eru e4urP erec {egY

T

(e

L

hrl -=.? 't=l l=.?l0 €l

:rcuqe81e rdueueldruoc

(r r r)*p1nr1n3'l L I l=r',:r.ur^v'ElrqBsro^urelso u lrop

.z =(r)opeurlqoes (q

l.o t t)

[,: ';)= ", q* = u esre^,r Be,rr,tu,,, '[,1 ';)= o arsa prcunrpe sorrr]er^r

(z r)'z="9'E-=t'g'l-= z:q'z-t'g :rcrrqe8lerdueueyduo3

[, ,)=, ,gc eutiqo eS 'EIIqBsre^uI erse V \xap 1= t-2.7 = (p,)tep rue.,ry (e

alinJog

(t e) :l l=v\I Z)

(too) 'l r t rl=r (q

[r t t) 'I(e

'g

)

w:::. !"i;ijii w,;!,";/ t;*,p"se ;ii:i

1. Afla1i care dintre matricele urmdtoare este inversabili:

(t2) (23) (ti) ,=l:j, :.l, u=fr'; il\3 o) \4 6) \, t) [o z t) ls ts _to]

2. Aflali care dintre matrice sunt inversabile qi determinali inversele lor:

", f , ,..l, b) c :), .r[, _r\ ( i 2\

[s 4) \2 6) tl ,,J' o'[-, 'i'

= ' , ,'=(' -')J;r(r) (-3 z)'

101 i) . Calcul5m

I t .i

I ^^ lrol. = IJ. d., = -1. .l = *1,r -' It tl

l: tlr.r. a.. =l l-')

Ir rl

: -1 o)ILI 't'r r) I I

)l

2)

; j *" inversabild pentru

r e l. se obline cd

mt - 5m +2 > 0 cu solufia

55 se arate ci matricele

,', - .1 ) gi. astfel,

-,i r ;i (1.+A)-t=Ir-A.

(cosa sina)t)

[-rin, "o"o )'

(2.

8. Fie matric." , =

[?

(t I 0\ (t -2 7\ (3 -l t)ol r r r l; erlo t -21; r,rl-: s -2

I

[o r r,J [o o r,J [r -2 r)3. Afla{i valorile parametrului a real pentru care fiecare din matricele de mai jos este inversabili

gi pentru valorile gisite determina(i inversa matricei:

(a ,) ,.r, o*,), .,[l :, 1'l, ,,li 1 ;.], .,['l' ";' llu' [o t)' o'[, o )' ''[; : ;,J' ''[; ], ;) "l ; i ;l

(3 o o) (z I 3)

{. Fie matric rte, .t=lz , o l,

u =lo , , I

[t 2 t) [o o r.J

a) Arltati cd matricele sunt inversabile;

b) Calculali A-' ,B-', (m)-' 9i ardta{i ce: (.lA)-' = 3-t ' tr-t '

(t a+t 2 )5. Ar6ta{i c6matricea l =1 1 I b+l l este inversabildpentru orice a,b e,4. qi determinali

[r 1 t)inversa ei.

(t o -,) (t o ')6.Fiematricele:l=10 I 0l.B=10 101

[o o rJ [o o rJ

a) Calculati AB qi BA;

b) Determina{i inversa lui B '

(r7. Se considera matricele I = I ^

\0

30\I

4 I l. Si se calculeze determinantul inversei matricei date.

0 t)

o -r\ (1 z)

, , 1,, =

I , -1 l.

Sunt matricele AB qi BA inversabile?

' tr 0)

::.'. ftare Determinanli

i.

t1ueulurJalaOBereip,ru3 ep riplrun ed rrircrexe rS auelqor4 .p-IX e VSy-IJ - yJIIVNiIJVI I

mrrslu &=12 e3rutrued (q

!,-8' Is ,-u Iieuguereq @

F elscuterutS g= z etg '92

W utP g'Y eIecI.IlBIu old 'W,

(r, €)ppl0 t Zl=.Vpcel.'EZ

[o o IJ

e- ' ('f + f,)rep i

= ('1 + r-,')leP

"(E) tltr 3 fr secl4s{u eld 'ZZ

,ttlni =(Q'o)r-w Pc fulgrv

(r- ,\rierp:nl' 'l=g gxe1

\I I,/

,= (q'o)W eeclJlelu PP eS

alse / BocIJtrBtu Ec IislPJV

'.'It-yl= rY g) Ietqry

0 r)r f Ol=V uertJlEuetg -t

0 z)

'r-g Il€uluuoleC

zlrqesre^ul tr €eclJlelu elsg

-ZI)- z Il=u EerlrteruelJ

I

:.- z l)',,((r)x) $ePc1u3

=, Inl olIJolBA rieururreleq

Y =(s)x'(v)x P'f€rerv

= l' ?3cIJl€Iu gloplsuoc eS

're es,re,\ur rfeuruueleprspyrqus-re,\urolso gr+'7 eecrrlerupcrielgry ''o= rg lgJuJie-Irsper,leluo g oljl .r,I

'pllqssJelulelse'€7b +yd = ,_(rt-tr ) o_rurn.r1ued,y.:>b,d eurlureleposes (q

iTtu = ,v erec n.rtruod x > ll./ ourruJolop es es (e

(ot t \

I t ,)=' eoclrleul ereplsuor es '9I

(o , z\r-g Pcli€]Prv ''I-tr=s ls lf z r l=r/ olecrrleruerc .sI

[t z t)

PIrqsSJeAuI else

.8''-n t=

'.Y = ,-V gcputtls w

',-tr, 1ie1nc1e3\. - ,Y )Y er tlElPrv

tr :secrJlPru or{ .nl

IEor InJler.uered {eUV .I I

: Y eecr4eru elc.ol

= tr eocrrlEtu eld .6

(q

(e

(q

(e

'tz

(q

(e

'v7,

'@*'ll=,(v-'t):lc{errsuoueg (q :e1=,v pcrielp-ry (e

ig s-)I - l= r Erruleur orJ .cll'fr 8- )

(Z \ t-t

I i '* l=,, [0 t )'7=/r,p:pcpurrlS x)u,w elrrole^ lieuv. (t+u'u I-ru)

\ r,, , )=' Earr'uur'u orl 'Zl

'leoJ r uJsoJecuo

:rrc=(,rs+v,

(r r o)lo r ol:[o o z)

(* , ,lt > u'l x I- * l=v

eecrrleru erec rrlued rir

(e x z)

olu,l

t)

;)D

riegy' y: r.a

(;i:

'6I

(:

(;

ta

'81.(r o).v = ,V 'le

,)= ,, pc puttlS eleer ereurnu g ts

Z*ZT I-MLLt I lut

. (qI}BUV 'i

\Z

l1i.: = l..i-t=A'0 t)

-l = ,-{'.

18. Se considerd matricea ^=l: ]l qi mul1im ea G:{x(")/ s,x(a) = I,+a'A}'

[3 3)

a) Ardtali ca x(,t)'x(n)= x(a+b+4ab);

b) Determinali valorile lui a e i*, pentru care X (a) este inversabild'

c) calculali (x(r))-'.(t 2 -3)

19. Fiematricea l=11 2 -3lgipentruunarealfixatfie B=aA+I:'

[,24)a) Este matricea I inversabild?

b) Determina{i .B-1.

(2. o l)| , ol.Fie matricea A=10

[r o 2)

Aratali cd A2 =4A-3Ir;Aratali cd matricea I este inversabild qi determinali I ''

Seddmatrice a M(a,b)=(':u b,l,n,z,eq'

\ b a-b)(t I )

D""a B=[l -r)u'at*icd

M(a,b)=a]z+bB ei B':=21';

Ardtali cd M'\a.bt=.lr:* -U)

(o 1 1)

Fiematricea AeU.(R',,t=lt O ,lCalculali detAEi At qiardta{ic6:

[r r oJ

1

det(,4-' + /,) = -dst( I + 1,).

(r 0 0)

13. oura l' =l2 i o I o.,..rina1i matricea ,4.

[, 2 t)

24. Fie matricele A,B dinM, (R) astfel incat ^,

=(t, l'l ar*,*' AB + 1BA)-,

2s. Fie aeR simatricer. r=[::;; :"JJ ,:t::;; i::la) Determina{i A-t Si B-t ;

b) Pentru ce a e R matricea -B este inversa hi A?

,: . _j- este inversabild

;nind cd: det A = 2,

1- I

: r::r ersabild.

l.:: rnversabild gi

20.

a)

b)

21.

a)

b)

'r't

.., tr.ilc Determinanli

(ueuIuJalaOOIareip.r.u; ep !-ulpn ed rrircrexe rS etuelqor4 .€-IX B VSV.I:) * y)IJV1IEIVIAI

a$n1og

)=[: :) " [: :)tu e1tence eAIozoJ es PS

0 z)o ' l= *,qr plrnzeut ,-lI z)rt€)I

) Ll=n, oullqoes

r o t)(roo)l, t ol=treecu,rJttzt)= J elSs rerlencoellnloS

z )q-( r z-')!=r-.,lr-[z- | )r-

"*- [: l)=o u*o

a$n1og

.: : ,l) =-l::):ey{unce ellozoJ 3s PS

s-), I

= _Y :elsr tctiuttre

'9 'i= "9 '[: '^)=, ,\8 t)

:Arseccns uro^v .ptrq,Sre^ur elso y ,ep.t = 8-6 = (r)pp ,u,ito .s [t !J: , ,,, (q

(r r) (r r)(r o'\

l'-'.i=[,t,Jl,-r):*atsctatlcncaerinlos

(r'_ o,)=

, o,,, [.] ?)= ., ,,.,

gpunlpeBoJul?W .l=rrg,O=rzg,Z-=.rg.l=,,g lunsrcrrqe8lel4ue*e1arro3 [l l)=r,

rsr=(r.)rrp..,n[l o) (t r) (t o) -.. Ir o) \' ' ' [z t)=Y ""'t'^ [, )',1, ;)=r poptlnzes

[' ",)=n

"" (E

tl i)=('J Il (,:,,r"r"s .[r: e-)=,

r. r* [,t_

t;)=., ,t=2,s,8_=,,e ,r_=,

(q

{')=r? tl (p\€/ (z t)

.{'.' ' 'l= , [f r] (q\r v) (r t)

rY

alinlog

.(r z') (r r) .l l=l l.x \0 L) \0 t)

{r 'l= " [, ol\r El lr t)

:ol€oculsru olniencs $u,r1ozeg

flfYflf,turyru rrJ,vncm

nI-'relse ro BSroAur rs gpqesrazrur alse

' ,-(,r)= ,(,-Y) :gc {e1gry 'oleal oror.unu eluetuole nc rop Inurpro op glrqusro^ur oJuletu o tr olC .gZ

'plrq€sro^ur elsa gr pcsp retunu tS gcep plrqusre^ur etso tr gc rialsuoureg (q:trg = gtr pc rlelgry (r

'g +v : gy tgttn 1e;lse (;r) z4 > g,y all .17,

, .t, BsJolrlr riuuruuelep rS ppqesrea.ur

frt*t e ,, ) alse u BerrJrBr.u ecrielp.ry'y-=r'q'r, 'l eg iq+l Dq l=H

eerurur.uelJ .gz

[ ,, qD ,D+l)

(c

(e

't

L

Y-'I (q i yg= -y (e

(t r r)

l, , 'l=zBerrrl,uerc '62

fr r t)

(e

(r

:ec elEJe es ps

= _\' olsa rerfunce e{n1og

. .,.,. '[o t] = , ,rno

[z t)r-t_(r r'l 1.t z,-

'- =[o ,-.,J [o t

=

- :.h 'l-=r?'l-=

(r r) rLUI\V I l=ValJ\0 t-)

.;: ai r'l3tricea I este

r\ ersabrlA.

':. .eale. .{rdta(i cd:

I

I -- I

( -r 0\ (-t 1)i) t'. ,=[ ,' ,J or","ca det( A)=-t, ''=[o

,,J' 4 =1, 4, =0'

n,, =-r. 6,,=-t. r.=[], l,) ,' , =-(1, l,)=[,' ?) t"'*'"sistemuruieste

(t o\ f-l o\ (-t o)

"=[, ,.J [ ' ,.J=[-, t)

r) o,.u ,=[l 1). ""..cd det( A)=2.'n=(:;) ^.=(], ,'l ,'

, '=:(:, ;)Soru{ia ecuariei este x = i[r, ;) ii) = iti)

2, Sd se rezolve ecualiile:

(t 2\ (-t z :)-r) t,,,J "=[ , ;;),

Solufie(r z)

.r) o".a I = [i i)rr,"u^ cd det(l) =

,,_-t( I -2)=lf-, ,).' =T[-, rJ-][ 2 -t)

Solulia ecualieieste X = 1[ -'- 3\. 2

ot)t.1 t)

3) Matricea r=[i : i] ""

det(z)=l,deciesteinversab,d.

(t o o.1 (rSeobline 'o=1, ' o

I' o. =1,

[: t t) lo

(2 ,o)r, _,l-r , ,l.l o ,Rezutti"aX=l + ,,1[; ;[z o 1/'

3. Si se rezolve ecualia matriceal5:

(t ') , [3 2]-f r 'l[: s,] '^ \+ t) lo t)Solufie

2l-r 3

4020

(t z :)b) ,lo I rl=

[o o r.J

2\/-r z :) r(t o 3) (t-,.J[ ' , ,J=;[-, : :J=[-'

o\,l

t.1lt)

(ti-r

-2 -'), -, l.0 1)

-2 -l\ (r, -, I Ei a-'=l o

o t) [o('t -1 -1-r\ I -I l-l 5 -1-ll=l

, J l4 -8 -'3

' (2 -2 -l

-z)t.t)

t :\{= laredet(,1)=1q10 l/ \ /

'. D-:: = t. lututri."u adjunctd

rsacild. Avem succesiv:

'3 -t\i: .-{ ^ l. Solulia-8 3)

L1,10 Determinanli