LJBR MTRIKS Bil kit mempui sutu sistem persm liier + + z = + + z = + z = Mk koefisie tersebut di ts disebut MTRIKS, d secr umum dpt ditulisk sbb :
Jjr bilg tersebut di ts disebut MTRIKS, d secr umum dpt ditulisk sbb : i m i m i m ij j j ij mj i m m, dlh bilg bult ij = eleme-eleme dri mtriks (i =, m) (j =, ) m bk bris orde mtriks m bk kolom Mtriks bis ditulis deg otsi ()
Mcm mtriks Mtriks bujur sgkr, bil m = 7 8 9 m Eleme-eleme,,, disebut eleme-eleme digol utm
Mtriks bris, bil m = Mtriks kolom, bil = Mcm mtriks [ ] m [ ]
Mcm mtriks Mtriks ol, bil ij = :
TYPE MTRIKS BUJUR SNGKR Mtriks Digol, Jik semu eleme sm deg ol, keculi eleme-eleme digol utm ij = ii
Mtriks Stu (uit mtriks) Jik eleme-eleme digol sm deg d eleme-eleme g li sm deg ol Disebut jug mtriks idetits = [ I ] = [ I ] TYPE MTRIKS BUJUR SNGKR
Mtriks simetris, jik ij = ji Mtriks skew-simetris, jik ij = - ji 7 7 7 7 TYPE MTRIKS BUJUR SNGKR
OPERSI MTRIKS Kesm mtriks Du mtriks [] d [B] diktk sm bil ij = b ij [ ] d [ B ] hrus mempui orde g sm
OPERSI MTRIKS Pejumlh mtriks Bil [] d [B] pu orde g sm, mk kedu mtriks tersebut bis dijumlhk mejdi mtriks [C] [C] = [] + [B] c ij = ij + b ij Sift-sift pejumlh Mtriks [ ] + [ B ] = [ B ] + [ ] Komuttif [ ] + [ B ] + [ C ] = ([ ] + [ B ]) + [ C ] ssositif
[C] = [] = [B] = EXMPLE : 9 7 [C] =
OPERSI MTRIKS Perkli deg sklr : Sutu mtriks [] dpt diklik deg bilsklr k meghsilk sutu mtriks [D] = k [] d ij = k ij Sift-sift perkli sklr mtriks: k ( [] + [B] ) = k [] + k [B] k ( [] + [B] ) = ( [] + [B] ) k
EXMPLE : [ ] = ; k = - [ D ] = 8
OPERSI MTRIKS Perkli mtriks Mtriks [] mp d [B] p dpt diklik meghsilk mtriks bru [E] m = [] mp [B] p e ij k b dim : i =,, m ; j =,, ; k =,, p p ik kj
) ( ) ( [] = ; [B] = [E] = [E] = EXMPLE :
Sift-sift perkli mtriks : [] ( [B] + [C] ) = [] [B] + [] [C] ; sift distributif ( [] + [B] ) + [C] = [] [B] + [] [C] ; sift distributif [] ( [B] [C] ) = ( [] [B] ) [C] ; sift ssositif [] [B] [B] [] [] [B] = [] [C] ; belum tetu [B] = [C]
TRNSPOSE MTRIKS Jik mtriks [] deg orde m Trspose mtriks [] = [] T dlh mtriks berorde m deg bris d kolom mtriks [] mejdi kolom d bris mtri [] T [] = [] T = EXMPLE :
Sift-sift dri trspose mtriks ( [] T ) T ( k [] ) T = [] = k [] T ( [] + [B] ) T = [] T + [B] T ( [] [B] ) T = [B] T [] T
DETERMINN MTRIX BUJUR SNGKR [] = Det [] = [B] B b b b b b b b b b b b b b b b b b b b b b b b b Co-fctor b ij = ( ) i+j Mior b ij
Utuk mtriks deg orde g lebih tiggi ( ) cr sm k ik c ik i c i i c i i c i dim c ik = co-fctor ik
INVERS MTRIKS BUJUR SNGKR Mtriks tidk bis dibgi deg mtriks li Sebgi logi, diguk INVERSE dri mtriks tersebut pbil [] d [B] dlh mtriks bujur sgkr, d [] [B] = [I] = [B] [], mk mtriks [B] disebut iverse dri mtri [], d mtriks [] dlh iverse dri mtriks [B] Seljut [] disebut mtriks NON SINGULR Bil [] tidk pu iverse disebut mtriks SINGULR Iverse dri mtriks [] bis ditulis [] -
[] = ; [] - = [] [] - = = [ I ] Ctt : Utuk mecri iverse sutu mtri dpt dipki beberp metod, tr li : metode d-joit, metode pemish, metode Guss-Jord, metode Cholesk, dsb EXMPLE :
Metode Guss-Jord k dicri iversi dri mtriks [] Lgkh-lgkh g dilkuk : ) mbil mtriks stu [I] ) Deg cr opersi bris, ubhlh mtriks [] mejdi mtriks stu ) Proses ke- jug dilkuk pd mtriks [ I ], sehigg setelh proses selesi mtriks [ I ] telh berubh mejdi mtriks [] -
7 [] = 7 [] - = LNGKH KE- LNGKH KE- LNGKH KE- LNGKH KE- LNGKH KE- LNGKH KE- Selesi????? EXMPLE :
MTRIKS ORTHOGONL Sutu mtriks bujur sgkr [] disebut mtriks orthogol bil [] - = [] T [] [] T = [] [] - = [ I ]
EXMPLE : cos si si cos [] = [] T = cos si si cos [] - = cos si si cos Kre [] - = [] T mtriks [] disebut mtriks orthogol cos si [T] = si cos [T] - = cos si si cos [T] T = cos si si cos Kre [T] - = [T] T mtriks [T] disebut mtriks orthogol
TEORI DEKOMPOSISI MTRIKS Bil [] = sebuh mtri bujur sgkr mk mtriks tersebut dpt diekspresik dlm betuk : [] = [L] [U] = l l l l l l l l l l u u u i u u u u u u u [L] = lower trigle mtriks [U] = upper trigle mtriks
EXMPLE : 7 9 = 9 7 7 8 [] = [L] [U] pliksi pd solusi persm liier simult : [] {X} = [B] [L] [U] {X} = [B] misl [U] {X} = {Y} [L] {Y} = [B] {X} = [U] - {Y} {Y} = [L] - [B] dpt diperoleh tp iverse mtriks Sehigg : {X} = [U] - [L] - [B] dpt diperoleh tp iverse mtriks
SOLUSI PERSMN LINIER SIMULTN Persm Liier Simult deg buh bilg tk dikethui dpt ditulisk sbb : b b b b b b b = Secr mtriks ditulis, [] {X} = [B]
+ + z = + + z = + + z = z = [ ] { X } = [ B ] { X } = [] - [B] z = =?????? EXMPLE :
PRTISI MTRIKS Sutu mtriks bis diprtisik mejdi SUB-MTRIKS deg cr h megikutk beberp bris tu kolom dri mtriks sli tur-tur g dipki utuk megopersik mtriks prtisi persis sm deg megopersik mtriks bis dim ; =
B B B B B B B X B B B 7 B B 8 B B 7 8 ] [ 8 9 [B] EXMPLE : sehigg ;
BEBERP RUMUS KHUSUS [ ] = Mtriks bujur sgkr d simetris ; orde : ij [ B ] = Mtriks empt persegi pjg ; orde m : b ij { X } = Vektor kolom ; orde : i { Y } = Vektor kolom ; orde m ; i Bil ; T { X} [ ] i j ij i j Mk ; { X} [ ] { X} tu seblik ; { X} T X ; { X} [ ]{ X }
X } ]{ [ } { X X T = ½ ( ) EXMPLE : Ф Ф
8 } { } { X X
Y B X m m T ] [ } { m m Y B X Bil ; Mk ; 9 8 7 B { X } = { Y } = 9 8 7 Y B X T EXMPLE :
9 8 7 9 8 7 8 7 9 9 8 7 Y B X