SISTEM PERSAMAAN LINEAR ATAS RING KOMUTATIF. Titi Udjiani SRRM Jurusan Matematika FMIPA UNDIP Jl. Prof Soedarto, S.

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1 SISTEM PERSAMAAN LINEAR ATAS RING KOMUTATIF Titi Udjii SRRM Jurus Mtetik FMIPA UNDIP Jl. Prof Soedrto, S.H, Serg 575 Abstrct. Lier systes equtios over couttif rig re lier systes equtios with the coeffici-ets of these equtios re eleets fro couttif rig. This er discusses bout bsic theores o solutio of lier systes equtios over couttif rig. The bsic theores will be foud by usig chrcteristic of idel,ihiltor d rk of coefficiet trices of lier systes equtios over couttif rig. The idel is geerted by iors of coefficiet trices of lier systes equtios over couttif rig. Coutig the ihiltor of idel we get the rk of coefficiet trices of lier systes equtios over couttif rig. Key words: idel, ihiltor, rk.. PENDAHULUAN Slh stu slh etig dl tetik dlh eyelesik Siste ers lier. Kre lebih dri 75 % dri slh tetik yg dijui dl liksi ilih uu idustri elibtk eyelesi siste ers lier higg th tertetu. Deg egguk etode etode tetik oder, serigkli kit dt ereduksi sutu slh yg ruit ejdi siste ers liier. Siste liier serigkli ucul dl eer bidg bidg seerti erdgg, ekooi, sosil, ekologi, deogrfi, geetik, elektroik, tekik d fisik [4]. Oleh krey deg eliht kodisi dits k eh tu ebhs egei siste ers lier hrus terus eerus diuyk eigkty. Tulis ii ebhs slh stu jeis siste ers lier yitu siste ers liier deg koefisie koefisiey ggot dri rig kouttif. Pebhs dibtsi d eyelesi tu solusi dri siste ers lier.. SISTEM PERSAMAAN LINEAR ATAS RING KOMUTATIF Diberik Siste ers lier sebgi berikut b b (.) b Siste ers lier (.) ereresetsik buh ers liier dl buh vribel tk dikethui,,...,. Koefisie ij d kostt b i dlh elee elee dri rig kouttif R. Siste ers lier (.) dt diytk dl betuk triks AX B (.) deg A ( ij ) M (R) B (b,b,...,b ) t R X (,,..., ) t R Pers (.) tu (.) diktk euyi solusi di R, jik d vektor ξ R sedeiki sehigg Aξ B. Jik B O, k siste ers lier A X O disebut siste ers lier hooge. Siste ers lier hooge sellu euyi solusi, lig sedikit euyi stu solusi yitu ξ O (,,...,) t R. Solusi ξ O disebut solusi trivil dri AXO. Solusi ξ R disebut solusi o trivil dri AXO jik ξ O d Aξ O. Pebhs ert k ebhs egei teore terkel dri N.Mc.Coy egei syrt erlu d cuku siste 7

2 Jurl Mtetik Vol. 9, No.3, Deseber 6:7-33 ers lier hooge AXO euyi solusi o trivil. Teore. ( Th. N.Mc.Coy) [4] Dikethui A M (R) Siste ers lier Hooge AX O euyi solusi o trivil jik d hy jik rk (A). Dikethui siste ers lier hooge AXO euyi solusi o trivil rtiy jik ξ R dlh solusi dri Siste ers lier hooge k ξ O. Berrti bhw terdt koordit driξ isly [ξ ] k (.3) * Jik. Meurut teore 4. [4], dikethui bhw rk (A) i {,}. Sehigg kre k rk (A). * Jik Misl ( i,...,i ;,...,) dlh ior berukur () dri A. Dibil triks erutsi P Gl(,R) sedeiki sehigg PA euyi bris bris i,...,i dri triks A sebgi bris erty. Jdi Row (PA) Row i (A), Row (PA) Row i (A), Row (PA) Row i (A). Sehigg PA dt diytk sebgi berikut. PA i i i i i i i * berukur,. Keudi dibetuk i i i i i i i i D i i i dlh keugki-keugki dri bris A d det (D) ( i,...,i ;,...,). Dikethui bhw A ξ O d D dlh keugki keugki dri bris A k Dξ O. Seljuty ξ det (D) ξ. Berdsrk (teore.,[4]) bhw det(d) (dj D) D k ξ (dj D) Dξ. Sehigg ξ O. Berrti utuk seti k dri ξ berlku [ξ ] k O (.4) Seljuty kre ( i,...,i ;,...,) dlh sebrg ior berukur () dri A d egguk ers (.4) k [ξ ] k A R (I (A)). Berdsrk (.3) [ξ ] k k A R (I (A)) (). Akibty deg egguk (defiisi 4.,[4]) bhw rk(a) ks {t AR It(A) )()}, k rk(a). Seljuty jik dikethui rk(a). Misly rk(a) r. Jik r k kit dt ebh beber ers deg koefisie ol d ers (.), sehigg dieroleh siste ers lier hooge yg bru sebgi berikut. (.5) Utuk ebuktik bhw Siste ers lier hooge AXO euyi solusi o trivil dt dibuktik deg eujukk bhw siste ers lier hooge (.5) euyi solusi 8

3 Titi Udjii SRRM (Sisti Pers Liier ts Rig Kouttif) o trivil. Artiy terdt ξ O R sedeiki sehigg eeuhi ers (.5). A Kre I t (( ) I t (A), deg O dlh O triks Nol berukur, -, t Z ; A k rk(a) rk ( ). O Jdi deg eggti ers (.) deg ers (.5) d berdsrk d (teore 4.,[4]), tet dt disusik bhw r i {,}. Kre rk(a)r k A R (I r+ (A)) (). Dibil d A R (I r+ (A)). Jik r k A R (I (A)). Dili ihk ξ (,..., ) t R dlh solusi o trivil dri AXO. Jdi dt disusik bhw r i{, }. Kre rk(a)r d rk(a)ks{ t AR (It (A)) ()} k A R (I r (A))(). Artiy d ior (rr) dri A yitu ( i,...,i r ;j,...,j r ) dri A sedeiki sehigg. Kit dt eukr tu eggti bris bris i,...,i r d kolo kolo j,...,j r dri A ejdi r bris ert d r kolo ert dri triks bru (.5), deg eglik d sisi kiri d k dri A deg triks erutsi P Gl(,R) d Q Gl(,R) sedeiki sehigg C * PAQ deg C M rr (R) d * * det (C) ( i,...,i r ;j,...,j r ). Mislk ers (PAQ)XO euyi solusi o trivil β R k (PAQ) β O. Kre P dlh triks ivertible k P - (PAQ)β O; A(Qβ ) O; Aξ O. Seljuty kre ξ β Q, seetr β O d P Gl(,R) k ξ O. Terbukti bhw AX O euyi solusi o trivil ξ. Akibt. [4] Sutu siste ers liier hooge euyi solusi o trivil jik byky ers lebih kecil dri d byky vrbel yg tidk dikethui. Dibil siste ers liier hooge AXO deg byky ers lebih kecil dibdigk byky vribel yg k dicri. Artiy jik A M (R) k <. Deg egguk (teore 4.,[4]) k rk(a) r i{,} <. Seljuty deg egguk Teore terbukti bhw siste ers liier hooge euyi solusi o trivil. Teore.3 [4] Jik A M (R) deg det (A) U(R) deg U(R) dlh hiu elee elee uit di R, k utuk sutu B (b,..., b ) t R, ers AXB euyi solusi tuggl ξ (y,..., y ) t deg y j (det(a)) - b j j b utuk seti j,,...,. Dibil ξ (y,..., y ) t degy j (det(a)) - b j j b j + j + j + j + utuk seti j,,...,. Kre det (A) U(R) k det(a) ivertible sehigg det(a)y j b j j b j + j + 9

4 Jurl Mtetik Vol. 9, No.3, Deseber 6:7-33 j b j + j b j + b i cofij (A) i y cofi(a)bi i dj(a) B y cofi(a)bi i det(a) ξ dj(a) B Kre det(a)idj(a)a k dj(a) A ξ dj(a) B. Kre det(a) U(R) k dj(a) ivertible, sehigg A ξ B. Seljuty k dibuktik bhw ξ tuggl. Didik ξ ' dlh solusi li dri AXB, k Aξ ' Aξ B; A(ξ ' - ξ )O. Kre A ivertible k (ξ ' - ξ) O d ξ ' ξ. Dri teore dits dt disiulk bhw jik A ivertible k AXB euyi solusi tuggl utuk sutu B R. Teore berikut k ejelsk syrt erlu sutu Siste ers lier AXB euyi solusi deg A (R) d B R. Teore.4 [4] Dikethui A M (R). Jik AXB euyi solusi k I t ( A B)I t (A), utuk seti t Z. Jik k kit dt ebh vribel vribel bru +,..., deg koefisie ol d ers (.), sehigg dieroleh Siste ers lier bru A ' X ' B sebgi berikut. b (.6) b Sehigg ξ (y,..., y ) t R diktk solusi dri ers (.) jik d hy jik ξ ' (y,..., y,,...,) t R dlh solusi dri ers (6), d utuk seti t Z berlku I t (A) I t (A ' ) d I t ( A B) I t ( A ' B ). Oleh krey jik I t ( A ' B ) I t (A ' ) utuk seti t Z k I t ( A B) I t (A). Artiy deg egguk ers (6), secr s t egurgi keuuy berlku jug utuk. Kre A d ( A B) s s euyi bris,k I t (A) I t ( A B) () jik t i {,}, sehigg kit dt egsusik bhw t i{, } Meurut defiisi I t (A) I t ( A B). Seljuty tiggl dibukti bhw I t ( A B) I t (A). Dibil (i,...,i t ;j,...,j t-, +)ior berukur tt dri ( A B) yg eut B d kolo terkhiry deg i <... <i t d j <... <j t- I t ( A B) Ak dibuktik bhw ( i,...,i t ;j,...,j t-,+) I t (A) Misl ξ (,..., ) t R dlh solusi dri AX B. Deg egguk (defiisi.,[4]) dieroleh Col (A) Col (A) B di R, sehigg ( i,...,i t ;j,...,j t-, +) i j i j i t j i jt i j i t j t i i i i i i t t 3

5 Titi Udjii SRRM (Sisti Pers Liier ts Rig Kouttif) i j i j k k i t jt i jt i jt it jt i k ik I t (A). i t jk Teore.5 [4] Dikethui A M (R) deg, rk (A) d B R. Jik terdt idel U di R d sebuh elee reguler z R sedeiki sehigg U I ( A B ) * R z U I (A) k siste ers lier AX B euyi solusi. Deg I ( A B ) * dlh idel di R yg dibgu oleh ior ior berukur dri (A B) yg eut kolo B. Kre rk(a) k A R (I (A)) (). Deg kt li I (A) (). Jdi terdt ior berukur dri A yg. Misl ( i,..., ;j,...,j ) dlh ior () dri A deg j <... <j.perhtik subtriks berukur dri A sebgi berikut. j j A k det ( A ) j j d b b b det( A ) ( A )(dj A ). b b b (.7) Dili ihk b (dj A ) b Seljuty dibetuk b jcofj( A ) j (.8) b jcofj( A) j c b jcofj( A ) j R. c b jcofj( A) j Sehigg utuk seti i,,.., berlku c i b j cof ji ( A ) j ij j b b i j j I ( A B ) * (.9) Deg egguk ers (7),(8), d (9) dieroleh : b j b j c b j b j c Deg b i ij u c u, utuk seti i u,,...,. Seljuty y i,..., y didefiisik sebgi berikut : { } { } ;v,... j,..., j y v I ( A B ) *, ci ;v ji ;i,,..., utuk seti v,...,. Sehigg iv yv ij y j ij y j ij c ij c b i, utuk seti i,...,. Jdi b i utuk seti i iv y v,..., d y i,..., y I ( A B ) *. Misl,... dlh ior ior berukur dri A yg. Mk utuk seti k,..., terdt sklr {y kv R, v,...} I ( A B ) * sedeiki sehigg k b i iv y kv i,...,. (.) 3

6 Jurl Mtetik Vol. 9, No.3, Deseber 6:7-33 Sudh dikethui bhw U I ( A B ) * R z UI (A) d kre,... ebgu idel I (A), k didt elee q k k reguler z k deg q,...q U. Seljuty ers (.) ejdi k iv q k y kv q k kbi k qk k zb i, i,...,. (.) Kre utuk seti v,..., berlku k k q k y kv UI ( A B iv y kv ) * R z, k q k y k r v z, deg r v R. Sehigg ers (.) ejdi ivrv z z iv r v zb i, i,...,. Kre z elee reguler dri R k iv r v b i, i,...,. Deg kt li ξ (r,..., r ) t R dlh solusi dri AX B. Akibt.6 Jik A M (R) deg I (A) R k utuk sutu B R, siste ers lier AXB euyi solusi. Kre I (A) R k rk(a). Utuk sutu B R berlku RI ( A B) * R z RI (A) Kre dlh elee reguler dri R k deg egguk Teore.5 terbukti bhw Siste ers lier AX B euyi solusi. Cotoh Dikethui siste ers lier o hooge deg koefisie koefisie ggot rig kouttif R Z/4Z sebgi berikut. + y + z + y + z + z 3 Ak diselidiki kh siste ers lier o hooge dits euyi eyelesi. Dri siste ers lier o hooge dits dieroleh A M 33 (R) d B R 3 3 deg R Z/4Z. Elee d 3 dlh elee reguler dri R Z/4Z R { r i r i Z/4Z } Z/4Z R { r i 3 r i Z/4Z } Z/4Z I 3 (A)Idel yg dibgu oleh ior ior berukur 33 { A R (I 3 (A)) { Z/4Z seti I 3 (A)} (). Rk A 3. A ) * { I 3 ( B 3 Z} {Z/4Z}, utuk + 3 3,,, 3 Z} Z/4Z 3 Dibil U Z/4Z d elee reguler k berlku U I 3 ( A B) * R U I 3 (A). Seljuty deg egguk Teore.5 terbukti bhw siste ers lier dits euyi solusi. A 3 U (R) ; ( A ) - 3. y ( A )

7 Titi Udjii SRRM (Sisti Pers Liier ts Rig Kouttif) y ( A ) y 3 ( A ) y 3 ξ y dlh eyelesi dri y 3 siste ers lier dits. 3. PENUTUP Kovers dri Teore.4 tidk berlku utuk rig kouttif R rtiy jik I t ( A B )I t (A), utuk seti t Z, tidk eji bhw siste ers lier euyi solusi. Ak teti jik R dlh field k kovers dri Teore.4 berlku, sebb jik I t ( A B ) I t (A), utuk seti t Z, k rk(a B) rk(a). D berdsrk [] k siste ers liier euyi eyelesi rk(a B) rk(a). 4. DAFTAR PUSTAKA []. Howrd Ato, lih bhs oleh Ptur Silb (995), Aljbr Liier eleeter, Erlgg. []. htt:/e.wikiedi.ovg/wiki/couttive_rig, dikses terkhir tggl 8 Deseber 6. [3]. Steve J. Leos (998), Lier Algebr with Alictios Pretice- Hll, Ic. [4]. Willi C. Brow (993), Mtrices over Couttive Rigs, Mrcel Dekker, Ic. New York. [5]. siulteous.df, Solvig Syste of Lier Equtio, dikses terkhir tggl 8 Deseber 6. 33