Matematike 1

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<ol><li> 1. MATEMATIK 1 Detyrat zgjidhura pregaditje prprovim Nga: ErmonCervadiku [UNIVERSITETI ILIRIA] </li><li> 2. 2 Elementet e algjebrs lineare-Matricat Veprimet me matrica Mbledhja e matricave 1. Jan dhn matricat? 21 10 A , 10 02 B , 21 01 C Njehsoni: b) 2A-2B+C Zgjidhje: 1. b) 21 01 20 04 42 20 21 01 10 02 2 21 10 222 CBA 03 23 . -------------------------------------------------------------------------------------------------------------------------------------------- 2. T kompletohet matrica vijuese? Nse jan dhn a21 = 4, a32 = 5, a13 = 3, a23 = 6, a12 = 7, a31 = -2. A = 9__ _8_ __6 Zgjidhje: 2. 952 684 376 A . -------------------------------------------------------------------------------------------------------------------------------------------- 3. Gjeni vlerat e ndryshme x,y,z,t q t vrtetojn barazimin? b) . 32 4 21 42 t yx t x tz yx Zgjidhje: 3. b) 32 4 21 42 t yx t x tz yx 321 442 tt yxx tz yx Pr ta lehtsuar mnyrn e zgjidhjes s problemit, prej shprehjes s siprme marrim: I) 442 xxx ; II) 002444 yyyyyxy ; III) 3332 tttt ; IV) 2311 zztz . Pra, parametrat e krkuar q e vrtetojn barazimin jan: ;4x ;0y ;2z ;3t Prova: 32 4 21 42 t yx t x tz yx 32 08 32 08 332 044 )3(21 44 32 042 ; -------------------------------------------------------------------------------------------------------------------------------------------- </li><li> 3. 3 Shumzimi i matricave: 4. Matrica A sht e tipit 2x5. Sa duhet t jet numri i shtyllave t matrics B, q t ekzistoj prodhimi AB . Po numri i rreshtave t matrics C q t ekzistoj prodhimi CA .? Zgjidhje: 4. Numri i shtyllave t matrics B duhet t jet 2, ndrsa numri i rreshtave t matrics C duhet t jet 5. -------------------------------------------------------------------------------------------------------------------------------------------- 5. Tregoni cilat shumzime jan t mundshme dhe gjeni matricn e prodhimit? a) ; 3 2 1 651 431 e) 5 0 3 2 ; f) 210 411 302 210 ; h) m m n m 00 01 1 0 . Zgjidhje: 5. a) 3 2 1 651 431 matrica e par sht e tipit (2x3), ndrsa matrica e dyt sht e tipit (3x1), meq numri i shtyllave t matrics s par sht i barabart me numrin e rreshtave t matrics s dyt ather ekziston mundsia e shumzimit t ktyre matricave. Matrica e prodhimit sht 3 2 1 651 431 29 19 18101 1261 , matric e tipit (2x1) e) 5 0 3 2 matrica e par sht e tipit (2x1), ndrsa matrica e dyt sht e tipit (2x1), meq numri i shtyllave t matrics s par nuk sht i barabart me numrin e rreshtave t matrics s dyt ather nuk ekziston as mundsia e shumzimit t ktyre matricave. f) 210 411 302 210 matrica e par sht e tipit (1x3), ndrsa matrica e dyt sht e tipit (3x3), meq numri i shtyllave t matrics s par sht i barabart numrin e rreshtave t matrics s dyt ather ekziston mundsia e shumzimit t ktyre matricave. Dhe prodhimi i ktyre matricave sht: 210 411 302 210 831440210010 Pra, matrica e prodhimit sht 831 , nj matric e tipit (1x3). h) m m n m 00 01 1 0 matrica e par sht e tipit (2x2), ndrsa matrica e dyt sht e tipit (2x3), meq numri i shtyllave t matrics s par sht i barabart numrin e rreshtave t matrics s dyt ather ekziston mundsia e shumzimit t ktyre matricave. Dhe prodhimi i ktyre matricave sht: m m n m 00 01 1 0 mnmn mm mnmn mm 0 000 0000 22 , matric e tipit (2x3). -------------------------------------------------------------------------------------------------------------------------------------------- </li><li> 4. 4 Detyra t kombinuara: 6. Kryeni veprimet? a) 33 75 12 34 51 413 120 ;d) 3 01 11 . Zgjidhje: 6. a) 33 75 12 34 51 413 120 4315843 160280 33 75 227 510 + 33 75 1910 25 . 6. d) 3 01 11 01 11 01 11 01 11 = 01 11 11 10 01 11 0101 0111 01 11 00)1(110)1(1 01)1()1(11)1(1 01)1(111)1(1 01)1(011)1(0 10 01 0111 0010 . -------------------------------------------------------------------------------------------------------------------------------------------- 7. Jan dhn matricat? 23 10 A dhe 12 11 B ; b) Shikoni a vlen barazimi: (A+B)2 = A2 + B2. Zgjidhje: 7. b) 222 BABA I) 15 21 15 21 15 21 1223 1110 12 11 23 10 222 2 BA 1110 411 11055 22101 II) 12 11 12 11 23 10 23 10 12 11 23 10 22 22 BA 106 26 30 03 76 23 1222 1121 4360 2030 . 1110 4112 BA dhe 106 2622 BA Nga kjo rrjedh q 222 BABA . -------------------------------------------------------------------------------------------------------------------------------------------- </li><li> 5. 5 8. Jan dhn matricat:? 12 11 A dhe 12 01 B ; c) Tregoni se: ))((22 BABABA . Zgjidhje: 8. c) ))((22 BABABA I) 12 01 12 01 12 11 12 11 12 01 12 11 22 22 BA . 24 02 14 01 10 01 14 01 10 01 1022 0001 1222 1121 . II) ))(( BABA 04 12 20 10 12 01 12 11 12 01 12 11 08 04 0080 0040 . 22 BA = 24 02 dhe ))(( BABA = 08 04 Nga kjo rrjedh q ))((22 BABABA . -------------------------------------------------------------------------------------------------------------------------------------------- 9. Njehsoni prcaktort e rendit t dyt? a) 14 23 ; e) 11 12 a aa ; f) bb aa 1 . Zgjidhje: 9. a) .583)24(13 14 23 e) 111111 11 1 22222 2 aaaaaaaaa a aa f) babbababba bb aa ))((1 1 -------------------------------------------------------------------------------------------------------------------------------------------- </li><li> 6. 6 10. Njehsoni prcaktort e rendit t tret nprmjet plotsve algjebrik? 1). T rreshtit t par 2). T rreshtit t dyt 3). T shtylls s tret b) 121 240 123 Zgjidhje: 10 b) Sipas rreshtit t par: 21 40 )1(1 11 20 )1(2 12 24 )1(3 121 240 123 312111 84440202443 . Sipas rreshtit t dyt: 21 23 )1(2 11 13 )1(4 12 12 )1(0 121 240 123 322212 8168)26(2)13(40 . Sipas shtylls s tret: 121 240 123 40 23 )1(1 21 23 )1(2 21 40 )1(1 333231 81216401226240 . -------------------------------------------------------------------------------------------------------------------------------------------- 11. Njehsoni vlern e prcaktorit: b) 721 045 132 . Zgjidhje: 11. b) 721 045 132 = 721 045 132 21 45 32 )357()2(02())1()4(1(2511037)4(2 631050410056 . -------------------------------------------------------------------------------------------------------------------------------------------- </li><li> 7. 7 12. Duke u mbshtetur n zbrthimin e prcaktorit sipas plotsve algjebrik t nj rreshti ose shtylle, njehsoni prcaktorin? b) 10312 31012 10140 12101 . Zgjidhje: 12. Matrics t dhn n detyr nuk mund tia gjejm prcaktorin meq nuk sht katrore. -------------------------------------------------------------------------------------------------------------------------------------------- 13. Duke zbatuar vetit e prcaktorve njehsoni vlern e prcaktorve: b) 530 210 420 ; d) zyx ztx 333 321 . Zgjidhje: 13. b) Nse shtylla apo rreshti i nj shtylle sht e barabart me 0 ather prcaktori i asaj matrice sht i barabart me 0. Shtylla e par e matrics 530 210 420 sht 0 prandaj prcaktori i ksaj matrice sht 0. -------------------------------------------------------------------------------------------------------------------------------------------- Matricat inverse dhe ekuacionet matricore 14. Gjeni matricn e adjunguar t ktyre matricave? c) 51 32 ; d) 202 151 013 ; Zgjidhje: 17. c) 51 32 511 A 3)3(21 A 1)1(12 A 222 A 21 35 adjA . d) 202 151 013 10102511 A 2)0021(21 A 10)5(1131 A 0)22()1221(12 A 6022322 A 3)0113(32 A 10)5(20113 A 2)1203(23 A 16115333 A 16210 360 1210 adjA . -------------------------------------------------------------------------------------------------------------------------------------------- </li><li> 8. 8 18. a) 53 12 ; c) 54 13 ; e) 714 320 153 ; Zgjidhje: 18. a) 53 12 13310)13(52 53 12 det A 511 A 121 A 312 A 222 A 23 15 adjA adjA A A det 11 3 2 13 3 13 1 13 5 23 15 13 11 A Prova: 10 011 AA 10 01 13 13 13 0 13 0 13 13 13 10 13 3 13 15 13 15 13 2 13 2 13 3 13 10 3 2 13 3 13 1 13 5 53 121 AA . c) 54 13 19415 54 13 det A 511 A 121 A 412 A 322 A 34 15 adjA 19 3 19 4 19 1 19 5 34 15 19 11 A Prova: 10 011 AA </li><li> 9. 9 10 01 19 19 0 0 19 19 19 15 19 4 19 12 19 12 19 5 19 5 19 4 19 15 54 13 19 3 19 4 19 1 19 5 1 AA . e) 714 320 153 35860338125)314(3 714 320 153 det A 1131411 A 3621 A 1731 A 1212 A 1722 A 932 A 813 A 2323 A 633 A 6238 91712 173611 adjA 35 6 35 23 35 8 35 9 35 17 35 12 35 17 35 36 35 11 6238 91712 173611 35 11 A Prova: 100 010 001 11 AAAA 35 6 35 23 35 8 35 9 35 17 35 12 25 17 35 36 35 11 1 AA 714 320 153 = 35 42 35 69 35 8 35 6 35 46 35 40 35 24 35 24 35 63 35 51 35 12 35 9 35 34 35 60 35 36 35 36 35 119 35 108 35 11 35 17 35 72 35 55 35 68 35 33 . 100 010 001 35 35 35 0 35 0 35 0 35 35 35 0 35 0 35 0 35 35 -------------------------------------------------------------------------------------------------------------------------------------------- </li><li> 10. 10 19. Njehsoni 2 A nse: . 213 201 112 A Zgjidhje: 19. 213 201 112 A 11212 AAAA 12416 13 01 12 213 201 112 det A 22011 A 1)12(21 A 231 A 4)62(12 A 13422 A 3)14(32 A 113 A 1)32(23 A 133 A 111 314 212 111 314 212 1 11 A 11 AA 111 314 212 111 314 212 = 132111142 338314348 234212244 = . 011 867 332 -------------------------------------------------------------------------------------------------------------------------------------------- 20. Njehsoni 12 5)( AAAAf nse: . 325 436 752 A Zgjidhje: 20. 12 5)( AAAAf . 325 436 752 A I) 2 A 325 436 752 325 436 752 = 98356625151210 1212428930201812 21201414151035304 2 A * 362517 423150 131169 </li><li> 11. 11 II) -5A= ** 151025 201530 352510 325 436 752 5 III) adjA A A det 11 190161058410018 25 36 52 325 436 752 det A 18911 A 1)1415(21 A 1212031 A 38201812 A 4135622 A 34)428(32 A 27151213 A 29)254(23 A 2430633 A 242927 344138 111 adjA *** 1 242927 344138 111 242927 344138 111 1 1 A Prova: 100 010 001 11 AAAA AA 1 = 242927 344138 111 325 436 752 = = 100 010 001 72116189488713512017454 1021642666812319017024676 347235562 12 5)( AAAAf = = 362517 423150 131169 + 151025 201530 352510 + 242927 344138 111 = 75615 125718 211560 . -------------------------------------------------------------------------------------------------------------------------------------------- </li><li> 12. 12 15. Zgjidhni ekuacionet? c) 2 3 32 13 xx x d) 0 12 142 xx x Zgjidhje: 15. c) 2 3 32 13 xx x 2 3 )32(3 xxx 2/ 2 3 96 2 xxx 31612 2 xx 031612 2 xx 24 2016 24 40016 24 14425616 2/1x 2 3 24 36 24 2016 1 x dhe 6 1 24 4 24 2016 2 x . d) 0 12 142 xx x 0)2()1)(4( 2 xxx 01222 01)1)(2(02 0]1)1)(2)[(2( 0)2()1)(2)(2( 2 1 xxxx xxx xxx xxxx . 2 33 2 33 2 33 2 1293 033 32 3/2 2 xx x xx Zgjidhje e vetme reale sht zgjidhja e par , prndryshe dy zgjidhjet e fundit jan komplekse. -------------------------------------------------------------------------------------------------------------------------------------------- </li><li> 13. 13 - Sistemet e ekuacioneve lineare - 1. Zgjidhni sistemin? a) 543 52 yx yx Zgjidhje: Zgjidhja e sistemit me metodn e Kramerit: a) 543 52 yx yx D= 1064 43 21 Dx= 101020 45 25 Dy= 20155 53 51 1 10 10 y x D D X ; 2 10 20 D D Y y ; b) Zgjidhja e sistemit prmes metods s Gausit: 543 52 yx yx Ekuacionin e par e shumzojm me numrin 3 dhe ekuacionin e fituar e mbledhim me ekuacionin e dyt. Pra, 543 1563 yx yx 22010 yy 145522 xxx c) Zgjidhja e sistemit prmes matrics: 543 1563 yx yx 5 5 43 21 y x Shumzojm nga ana e majt me 1 43 21 meq matrica 43 21 gjendet n ann e majt t y x ; prandaj, 1 43 21 5 5 43 21 y x **11 5 5 43 21 43 21 43 21 y x </li><li> 14. 14 10 1 10 3 5 1 5 2 13 24 10 1 43 21 1 , ngase det 43 21 sht: 102314 43 21 , adj 43 21 = 13 24 , ngase 411 a 221 a 312 a 122 a 10 1 10 3 5 1 5 2 10 1 10 3 10 2 10 4 13 24 10 1 43 21 1 10 1 10 3 5 1 5 2 43 21 y x = 10 1 10 3 5 1 5 2 ** 5 5 2 1 10 5 10 15 5 5 5 10 10 01 y x y x 1 x dhe 2y -------------------------------------------------------------------------------------------------------------------------------------------- 2. T caktohet vlera e parametrave a dhe b ashtu q sistemi: byx ayx 46 13 a) t ket vetm nj zgjidhje; b) t ket pa fund shum zgjidhje; c) t mos ket zgjidhje; Zgjidhje: 2. 13 ayx byx 46 a a D 612 46 3 ba b a Dx 4 4 1 63 6 13 b b Dy a ab D D X x 612 4 a b D D Y y 612 63 1. Q t ket vetm nj zgjidhje sistemi, duhet t plotsohet kushti: 0D , pr fardo Dx, Dy. meq 21260612612 aaaaD . 2. Q t ket pafund shum zgjidhje, duhet t plotsohet kushti: ,0D ,0xD ;0yD D=12+6a=0 dhe Dx=4+ab=0 Dy=3b-6=0 6a=-12 4-2b=0 3b=6 a=-2 -2b=-4 b=2 b=2; Pra, a=-2, dhe b=2 3. Q t mos ket zgjidhje sistemi, duhet t plotsohet kushti: D=0, dhe mjafton vetm njra Dx 0 , ose Dy 0 212606120 aaaD </li><li> 15. 15 242024040 bbbabDx 7. Me metodn e Gausit t zgjidhen kto sisteme ekuacionesh? b) 344 5323 224 321 321 321 xxx xxx xxx Zgjidhje: b) 344 5323 224 321 321 321 xxx xxx xxx 344 321 xxx 344 321 xxx 5323 321 xxx III/(-3)+II *** 32 41514 xx III/(-3) 912123 321 xxx 224 321 xxx III/(-4) +I * 32 101518 xx III/(-4) 1216164 321 xxx )1/(*** 41514 32 xx 101518 32 xx **** )1/( 64 2 x 2 3 2 x *** 3 415 2 3 14 x 171542115415)3(7 333 xxx 15 17 3 x 15 23 15 6845 15 68 3 15 68 633 15 17 4 2 3 4 11111 xxxxx ------------------------------------------------------------------------------------...</li></ol>