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'il'i'- -
EAD
EDeterm
Syarat determinan adalah mariksnya harus buiur sangkar.
Mencari determinan ordo 2x2:
A2"2: hllli r,*i]lAl = A(1,1, x A(2,2t - A(1'2) x A(2'l)
COIYTOH
^=[? 3]
lAl =9.8-5.4=72-20=52
36
DETER}I I IIA'{HAL 1
inan Matriks
USTTNG PROGRAIUT
ro Drll A(a3)80fOBI=110880rcBrI=IT08{0 nEAD A(I"J)E0 I|EXT .I00 mxr r?O DATA 9,S,{360 x = A(l,I) * A (&8),. ryl3) * A(8,1)9O I,PN|IIT 'I)UIIN,UIIIAIT UAXBII8 A = '':Xloo mrD
HASIL PROGRAITI
DETERMINAN MATRIKS A = 52
Mencari determinan ordo 3x3 ada dua cara:
l. Cara Samrs2. Cara Kofakor
Cara Sam.rs:
lAl = [A(1,1) x A(2,2) x A(3,3) +A(12) xA(2'3) x A(3'1)+ A(1,3) x A(2,1) x A(3,2) I - [ A(l'3) x A(2,21 x A(3'l)+ A(1,1) x A(2,3) x A(3,2) + A(1'2) x A(2'1) x A(3'3) ]
Cara Kofaktor:K(10 = (-l)r+i lvtl
lAl : A(1,1)xK(l,1) + A(1,2)xK(1,2) + A(1,3)xK(l,3)lAl : A(1,1)x IA(2,2)xA(3,3) -A(2.3)xA(3,2)]+ ,
A(1,2)x [ -A(2,1)xA(3,3) + A(2,3)xA(3,1) I +A(1,3)x I A(2,1)xA(3,2) - A(2,2)xA(3,1) I
lAl : + A(1,1)xA(2,2)xA(3,3) - A(1,1)xA(2,3)xA(3,2)- A(1,2) xA(2, I ) xA(3,3) + A( 1,2) xA(2,3) xA(3,1 )+ A(1,3)xA(2,1)xA(3,2) - A(1,3)xA(22)xA(3,1)
lRl = + A(1,1)xA(2,2)xA(3,3) + A(1,2)xA(2,3)xA(3,1)+ A(1,3)xA(2,1)xA(3,2) - A(tig)xA(2,2)xA(3,1)'- A(1,1)xA(2,3)xA(3,2) * A(1,2)xA(2,1)xA(3,3)
lAl : I A(1,1)xA(2,2)xA(3,3) + A(1,2)xA(2,3)xA(3,1)+ A(1,3)xA(2,1)xA(3,2) I - t A(1,3)xA(22)xA(3,1)+ A(1,1)xA(2,3)xA(3,2) + A(1,2)xA(2,1)xA(3,3) l
Cara pemecahan rumusan determinan ordo 3x3Misalkan lnl = X
X = [ A(1,1)xA(2,2)xA(3,3) + A(1,2)xA(2,3)xA(3,1)+ A(1,3)xA(2,1)xA(3,2) tr - t A(1,3)xA(2,2)xA(3,1)+ A(1,1)xA(2,3)xA(3,2) + A(1,2)xA(2,1)xA(3,3) l.
JikaX-C-D,maka:
C = A(1,1)xA(2,2)xA(3,3)A(1,2)xA(2,3)xA(3,1)A(1,3) xA(2,1)xA(3,2) +
c- C + A(lJxA(zJ+1)xA(3J+1)MN
A(1,3)xA(2,2)xA(3,1)A(1,1)xA(2,3)xA(3,2)A(1,2)xA(2,1) xA(3,3) +
D = D + A(1J+2)xA(2J+1)xA(3J)NM
Diambil:
Ag"3 =A(1,3) lA(23) |
A(3,3) J
r A(1,1) A(1,2)I e(z,r) Nz.z)L61g,t) A(3,2)
lAl :A(1,1) A(1.2) A(1,3)A(2,1), N2,?) A(2,3)A(3,1) A(32) A(3,3)
A(1,1) A(1.2)A(?,1) N2,2'A(3,1) A(3,2)
+++
A(1,1) A(1,2) A(1,3) KIl 1\ _A(2,1) A(2,21 A(2,3) rL\"''A(3,1) A(3,2) A(3,3)
lll:|,^litl, fli,'J, K(1'2) =
A(3,1) A(3,2) A(3,3)
ll,r:1, ltl?,Xlil]) K(,,3) = (-,)r*3
r rrr+r I Nz,z't A(2'3) I(-r, I n(g,z) A(3.3) I
+ I [A(2'2)xA(3,3)-A(2,3)xA(3,2)]A(2,2)xA(3,3) - A(2,3)xA(3,2)
t ttt.'2 | n(z,r) A(2,3) |(-r, I e(g,r) A(3,3) I
- I [A(2,1 )xA(3,3)-A(2,3)xA(3,1 )]-A(2,1)xA(3,3) + A(2,3)xA(3,1)
A(2,1) N2,21 I
A(3,1) A(3,2) I
D_
A(3,1) A(3,2) A(3,3)
38
= + I [A(2,1)xA(3,2)-A(2,2)xA(3'1)l= A(2,1)xA(3,2) = A(2,2)xA(3,1)
M=J*1N=J*2
I
,,lil
irllillll
:,lt!lii
PROGRAIUI FLOWC}IART
L
C = C + A(1J)xA(2,.tt{)xA(3,N)
D - D + A(1,N)XA(2JUI)XA(3J)
Ketenhtan:
- Jika'M : 4, maka M = I'
-JikaN=4,makaN=1-JikaN=5,makaN=2Perhih,rngan deterrninan dengan cara Samrs:
lAl =
lAl = (1xlx4 +2x4x4l:(4+30+
_-,-a
tz4132
3lal
23ts24
r1 2e: le I
L3 2
1
43
2x5x3 + lx4x2) - (3xlx3 + lx5x2 +
24)-(9+10+321
Perhitungan determinan dengan cara kofaktor:
= *1 (lx4 - 5x2)=4-10--6= -l (4x4 - 5x3)--l(16-15)=-l= +l (4x2 - lx3)=8-3=:,
12341t32r123415324123/l15324
Krr = (-l;t*t
Krz = (-l;t*z
Kr3 = (-l;t*s
lAl = A(1.llxK(l,1) + A(l'2)xK(lP) + A(l'3)xK(l'3)
lAl = t x (-6) + 2 x (-1) + 3 x (s)
=-$+-2+15-l
1524
4534
4132
I
40
0B I=t I0 3
0B J:l I0 3
()fi rl=1 I0 3
ll : J+l
ll : rI+?
4t
USTIIJG PROGRAIUI
lil,rllIt r i
I
i
tlitili
I0 LPBII{T CI{n$(16)AO N8U PROGEAM DBIEB,IIINA}I ON,DO 6X660 DrM A(s,8){0 LPBII|T 'XlAIRIf,8 A :"60FlBI=IT0g60KnJ=IIOU?O BEAD A(I"J,80 IIBnm IISIN0 "++ "S(LrI)i00 ll$m cI
rcO LPHM : LPRII\IT
110 ![EXT Iu0 DATA I,a,C,{,1,6,6,A4ICOFOBJ=1I05I40M=rtr*116'0tf=.1 +8160IFl[=4TI{EIIM=11?0 IF l{ = 4 llIElI N = I EISE IF N = B TTEN N = 2
180 C = C + A(I,ar).'r. A(e,[d) * A(g,N)190 D = D + A(I,N) * A(8,M),* A(C,J)800 NEXT e,
210X:C-D480 LIRIIIf "DEIEruIINAI{ A(6,6) = "1860 EI{D
IIASIL PROGRAITI
MATRTKS A :
't23
324:, ll'I.;!
DETERMINAN A(3,3.f = 7, .
Agar bersifat unirrersal, maka sebaiknya data dimasukkan melaluikeyboard (dengan statement INPUT).
USTIT{G PROGRAM
10 crsg0 ffPt T "ordo mstr{ko bqlu! 8an$a,r = ",N
42
50 Dru A(!I,N),P(!&P),[PG-a),Dr(A),m(}I-A)40 fOB I : I Ig N : FOB tI : I 1O N : PRIM't8rrs ";I;"
kolom "iJ;: INilI A(I,J) : NEXI <I : PBIM : MI$ I60 fOB I = I 1! N-8 : P(E)={F,F) : f,P(F)=E : f,B(F)=l : MXf f00 Df,(l)=Q : DK(8):0 : DK:I : lI0=1?0r080:1IOil-880 If f,P(O)=N0 TIIEN N0=N0*I : GOI0 70
90 I{EXT 0100 Ir Dtr<g m{EN DK(DK):NO : DK:DE*I : N0=N0*l : GOT0 70
1r0 DK:A(lr-r,DE(1)) '. A(N,DK(2)) - A(N-LDr(2)) * A(NpK(1))I8O TOB X = I TO N-8 : Df,=DK*P(X) : NEXT X
150 D=D*DK140 F08 F = N-8 T0 1 STEP -I : IP=KP(F)160 FP=f,P+I : II f,P>N IIIEN 410
100IrF=11II8N800170 FOB G = F-l T0 I SIBP -II80 IF (?=f,P(C) TIIEN 160
190 NEXT G
300 f,R(F)=f,fl,(I)+1 : Z\=E : ZL=W : GoSIIB 640 : GOIo 880
410 NEXT F : COl0 600
8e0 tr f = N-8 TIIEN 60 EISE N0:I&50 fOB H = F+] Tg N-3
840FOBrI:11!F860 IF XP(rI)=lrc mElI N0:N0*1 : COI0 ?40
860 NEXI eI
870 IIB(II)=I '. Zl=H: Z8=N0 : GOSID 630 : N0=N0+1880 NEXT H
890 G0r0 60
600 LPBII\IT "dotorminan = ";D610 END
680 KP(ZI)=@ : P(21):A(z1,zS) * ( 1)^(rB(z1)+r) : B,ETUBII
HASIL PROGRAM
ordo matriks bujur sangksr = 3barislkoloml?1barislkolom2?2barislkolom3?3
baris2koloml?4baris2kolom2?1baris2kolom3?5
43
baria3koloml?3baris3kolom2?2barie3kolom3?4
'i
determinan = 7
EAD
E
2. Operasi baris elementer yang disebut juga ellminasj aau* Jgd*.
[^,,,]5 [,"r^-']3. Matriks Mjoint
. 1..
Dalam hal ini ldta akan mempergungkan cara kedua, yaitu dengqn,
coI{ToH f i I
, :ti, l(-;, 1
Tent*anlhh inrlers. mat*s n n i, ,1., l, r,,,
lt 3lIz 2J
lnvers lUlatriks
Penyelesaian invels matriks secara analifts ada 6ga:Qor6, $ihl: ,:, .t,
1. Eliminasi bhsa dari
A.A-r=lr.
Acxe =
r1 0lo 1
L6 o
r1 0lo ILs o
A-1 =
00It_L-
,r]
I ol-L 2J
-ti Iz)
!4o
Dengan cara OBE ( A I Iz )OBE€ (Iz lA-l)
1 0 -1'l-3 1 3l-10 2ro -1.l1 3lo2J
Irr tz 13.Ilzr zz xlLsr s2 s3J
b2t(-21---------€
b12(-i)"
Ir Z
l,z 2
It iLo 1
t o I bl(l)I ----------------0 1l
rt+ 01 u2(2) .-i rJ
-
lq 3l.z z
tt ilr oLo I
I31
^-': [],I - 1l'l
-r zJ
Tentukan Invers matriks Ar'lol1J
COT{TOH12 0
As"s=13 ILl 0
Dengan cara OBE
OBE(Alls)--+(lelA-r)
o'lurtil
? l--*o'lb2l(-3)?l-o
Io
l00100
!2
0o
12 0 1ls 1 oLr o I
[r o tt3 l 0Lr o 1
t4 15 1624 25 2634 3s 36
11 t2 132t 22 2331 32 33
!2-llI
ioo-tL, I o-1 02
0-r]
1
I r o -rluzstS)l-ti 1 ol--| -1 0 2r
Cara penyelesaian inrrerse matriks dengan Program:Misall€n matriks yang diketahui:
v,\3x3 -
Di mana11 + Indeks dari element X(1,1)12 + lndeks dari element X(1,2)13 + lndeks dari element X(-1,3)
2l + Indeks dari element X(2,t)22 + Indeks dari element X(22'l23 + Indeks dari element X(2,3)fl + Indeks dari element X(3,1)32 + Indeks dari element X(3,2)33 + Indeks dari element X(3,3)
M=l!
-$2
!-iI
l'r olo lL1 o
rl olo 1
Lo o
11 0lo 1
Lo o
!2-ti
o
0 o'1b31(-1)1 0 l-------+o 1J
I I o o'lb3(2)| -ri r ol.-------------| -i o 1J
t3(-i)
l1
11
t2
1t
r3
t1
t4
l1
15
11
t6
lt2l -(21x 1l 22-(2lxl2', 23-(2r x 13) 24-(2txt4) 25-(21x 15) 25-(2lxt6)
31-(31x 1l 32-(31x 12) 33-(31x 13) 34-(31x 14) 3s-(31x l5) 36-{31x16)
46 47
roo'lo I olo o lJ
12 O l'l 12 o 1
A3*3=13 I ol.---..-ls r o"J J Lr o 1J L1 o I
t4 15, 1624 2t' 2634 35 36
11 t2 t32t 22 2U31
'2 33
Perhitungan secara Program:
X(l,l) : 2 X(1,2) = 0 X(1'3) = 1
X(2,1) : 3 X(2'2) = 1 X(2'3) = IX(3,1) : 1 X(32) = 0 X(3'3) = 1
Untuk M = 1
D=X(MJI)=X(1,1)=2 ,,
K : I -+ X(l,l) : X(1'1) I D = 212 = |R = 2.+ X(1'2) : X(1,2) / D = Ol2 : O
K = 3 + X(1,3) = X(13) / D : ll2 = O'5
K = 4 - x41,4),= ryl'4) I D = ll2 = o'5K = 5 + X(1,5) = X1,5) I D : Ol2 = O
K = 6 -+ X(l'6) = X(1,5) / D = Ol2 : O
li=2
M=3
Kesimpulan:
0rJ)-0.m)x(rlf.J)X(lJ) = X(lJ) - X(Llt) x X(Irl,J)
(B-baris)(K-kolom)
Irlalta:
x(8.1) = X(B,K) - XB,rrl) x X(M,K)
I
t4 15 1624 25 2634 35 36
lr t2 132t 22 2331
'2 33
o.t o olo I olo o 1J
rl o o.5lg r oLr o l
B=l) aPakahl B:lul?
rt=lJ 1t=t;B=2) aPakah
I- B=M?Irl=1) e*l
v"-+ keluar jalur
t+ C = X@Jf)
= {l,l) = 3
-0Jilta B = I
K=J
X(B,K) = X(B,K) X(B'Itl) X X(tftK)
X(B,K) = X(B,K) - C x X(II1'K)
K = I - X(2,1) ] x(3'tl _ : X
x(l'l)
:
K = 2 - x(22,
=\rr : : X I,,',, =t
ll-(l2x2l l?-(12x22', l3-(t2x23) l4-(12x24) l5-(12x25) r6-(12x26)
2l
22
22
2,23
22
24
?2
25
22
26
22
3l -(32x21) 32-(32x?2) 33-(32x231 34-(32x24) lE-/aawDEl lA-It2v2Al
r l -(l3x3t) l2-(13x32) r3-(t3x33) 14-(r3x34) l5-(13x35) l6-(t3x36)
21-(23x31) 22-(23x32) 23-(23x33) 24-(?3x34) 25-(23x35) 26-(23x36)
3l
33
32
33
33
33
v33
35
33
36
33
49
K : 3 + X(2,3) = X(2,3) - C x X(1,3)0 -3xO.5
K : 4 + X(zA, =..X(2,4) - C x X(1,4)= 0 -3x0.5
frK : 5 -+ X(2,5)
_: x(2,q)
_ 3 X f,r,u)',
K = 6 -- x(2,6) I x(3,e1 : : X f,r,.,
0.5-l.tI
= _1.5
= -1.5
:t
-0
t
ontuk l,tl : 2
D:XMJI):X(2,2)=1
K : 1 .-+ X(2,1) = X(2.1) I D :K:2 -+ X(2,2) = X(2,21/ D =K : 3 .+ X(2,3) = X(2,3) / D =K = .4 + X(2.41 : X(2,4) I D :K = 5 -+ X(2,5) : X(2,5) / D =K = 6 -+ X(2'5) = X(2'6) / D =
O/1 =O1/1=1
-f .5/1 = -1.5-l.5ll ='-1.5
1/1=1O/1 =0
t'ri,i,
I
ll, ll[
lM
o1ol1J
11 0lo 1Lo o
t=r\M=21
o.5 0 0l-1.5 1 0lo o rJ
r1 0lo rIr o
X(B,K) = X(B,K) -K:1+X(1,1)_=
K=2+X(1,2)_:
K : 3 -r X(1,3) :
K=4+X(1,4)_:
K=5+X(1,5)_:
K=6+X(1,6);
0.5-1.5
o.5
0.5 0-1.5 1-0.5 0
,t' +C=X(B,M): X(1,2) = 0
C x X(lttK)
x(1,1) - C x'X(2,1)I -OxO =t
x(1,2)-cxx(2,210 -0xl =0
X(1,3)-CxX(2i3)O.5 -Ox-1.5 =0.5
X(1,4)-CxX(2,4)0.5 -0x-1.5 =0.5
X(1,5)-CxX(2,5)0 -0x1 =O
X(1.6)-CxX(2.6)O .-Ox0 -0
0.5-1.5
0.5
apakahB=M?(1 +2)
B = 31 apahah
l- B:M?M=tJ ta+tl
0.5-1.5
o.t
X.(B.K)=X(B,K) -C xX(I[,K) 1
K = I + X(3,1) ::
*,?,r, _ : X 1,r,r, ; O
K:2-X(3,2): X3,Z) - C x X(1,2) .
=, O -1xO :QK = 3-+x(3,3)I *(i,rl
_ i X iljj,r, = 0.5
K = 4 - X(3,41" = X(3.4) - C x X(1,4)!G, 0 -lxO.5 =,-1.5
K = 5 --+ X(3,5) : O|,u, - t X il(r,u, = eK = 6.- x(3,6)
==
*9,:, : : X f,r,., = r00.lI Olo 1J
r1 0lo 1
Lo oo
.1o
Ir olo IL6 o
0.5-r.5,-o.5
o'lolrJ
0.5-1.5-o.5
51
,:,Im:ZlB:3'l
t =2]
apakahB:M?Q=2)
apakahB=M?'(3 *21
v+ keluar jalur
t ,4!
€ C = X(B,M)
= X(32) = 0
o o'l101o2J
:)-
Ir olo ILo o
$:
lvl -
o.5-1.5
I
o.5- 1.5
-1
apakahB=M?(l *3)
X(B,K) : X(B,K) - C x X(lu!'K)
K = I --+ X(3,1) ::
*,3,r, _ 3 X fr,1, : e
K = 2 -+ x(3,2) : O3,r, _ 3 X !r,r, _ oK = 3 -' X(3,3) : X(3,3) - C x X(2,3)
i = o.5 -ox-1.5 -o.qK = 4 - x(3.4)
:= *,1'3]u:
3 X l,;;, = _r.5
K = 5--+x(3,5)== *,3,u, _3 X fr,u, _ oK : 6 + X(3,6)
=: *(?,., : 3 X f,r,., : t
0.5-1.5
o.5
UnhrkM=3
p = x(IrlJt) = X(3,3) = .0.5
K = I --+ X(3,1) = X3,l) I D =K = 2 -+ X(3,2) = X3.2) lD =K = 3 -+ X(3,3) : X(3,3) / D =K=4+X(3,4):X3,4)/D=K=5+X(3,5):X(3,5)lD=K=6+X(3,6)=X(3,6)lD=
52
o.s o o]-1.s 1 0l-o.5 0 1J
rl olo IL6 o
X(B,K) : X(B,K) - C x XIul'K)
K = I + x(1,1): x(1,1) _ 3.Jf3'r, : t
K = 2--+ x(1,2) =:
*,1,r, : 3J af'r, _ oK = 3 + x(r,3):: *,1:3,
_ ilf,i'r, _ oK = 4 -+ x(r,4)
== ol:1, _ tJ X,:t, _ I
K = 5 + x(r.5) I x(l,sl _ 3Jf3'u, _ o
K = 6 + x(1,6) I *(l.el :|.J:gn, = _t
t+ C : X(B,M)
= X(13) = 0.5
t+ C = X(8,[!1)
= X(2,3) = -1.5
I O -r]-1.s I ol-102J
11 0 0lo 1 -1.5Lo o I
B = 2.l apakah)_e:M?
M=3) <z+g)
OlO.5 = O
O/O.5 = 00.5/O.5 = I
-0.5/0.5 = -t0/0.5 = O
llO.5 : 2
X(B,K) = X(B,K) - C x X(t['K)
K = 1 + X(2,1) : T'r, _ i_|.#r;rl _ o
K : 2 + x(2,21 f *(?'rl _ t_|.#r;rl _ I
3 + X(2,3) = X(2,3) - C x X(3,3): -1.5-(-1.5)x1:
4->X(2.4) = X(2,4) - C x X(3,4): -1.5- (-1.5) x -1 =
K-
l(=
K : 5 - X(2,5) : *,?,u, _ i_Li€;rl :
K = 6+x(2'6): *(3'e):l_;.#r"'tl =
I-3-1
LISTIIYG PROGRAJTI
IO I,PRIM CHR$(T6)
AO N8M PROGMM IIENCABI ITTVERS IIATRITE A!0 cls{O PBIIIT "MASI'XGAIV ORDO MATBIffB A :''80 Il{PIrT "JIIMLAII BARIS lfiTnilf8 A = ";I60 INRIT "JITULAH KOIOM MATtsIKS A = "il?0 E I <> .I IIIEN I080qI=I*?90 DIM x(InI)I00 FOB B =I 10 I110 X(B'B+I) = !UO I{EXT B
rCO PN|IUI 'XIA8I'ffiAN EIAMEN !/IAT?If,B A :''I{0F088=I19II00fOBf,=I10I100 PBIM'BARtrS ";B;"K0IOM "itr;170 Il{Pm x(B,K)I@ NEXT KI9O PruMAOO NEXT B
il
e10 LlEIllT "MAIliI[8 A ";TAB(10'tI);"tri[,ATniIBg SATT AN (IDEI{TITA8)"
E8O I,PBIM260FOBB=l.I0Ie40 IOBf,=lT0J280 LPRN? X(B,r),860 NEXT Ke70 LPBIM : LPEIMe80 ilErr B
390 IJBIM : LPBIMS00rcBM=I10 I510 DX = X(M,l[)SA0FOBK=1.!0J680 X(U,r) = X(U,r) / DX
340 MXT r660fOBB=lTOI600Il8:MTIIEN4l0c70 DX : x(B,l[)!80DOBK:1T0elgg0 X(B,tr) = X(B,f,) - Dx * X(M,K) " \-.,-'/4OO NEXT K
4EO NEXT I[{80 LPBIM "IIAIB,IBS SATUAII ( IDEM$AS )";TA8(16*I);
"IMIEBS I[ATBiItrB A''440 I.PBII{T
460fOnB=1T0I460F08f,=IT0tI
480 NUXT tr.490 LPBII\E : LPRIM ,'.800 l[Exr B ., ,i, IOIO EI{D
Contoh 1, matriks 2 x 2:
MASUKKAN ORDO MATRIKS A :
JUMLAH BARIS MATRIKS A = 7 2JUMLAH KOLOM MATRIKS A = 7 2MASUKKAN ELEMEN MATRIKS A :
BARISlKOLOMl?4BAR]SlKOLOM2?3
-3
o -1'lI 3lozJr 1 0 -1'l=l-s I 3lL-r o 2)
6-t
11 0 0lo 1 oLs o 1
55
BARIS2KOLOMl?2BARIS2KOLOM2?2
MATRIKS A ; MATRIKS SATUAN (IDENTITAS)
4.00 3.00 1.00 0.00
2.00 2.00 0.00 1.00
MATRIKS SATUAN ( IDENTITAS ) ; INVERS MATRIKS A
1.00 0.00 1.00 -1.500.00 1.00 -1.00 2.00
Contoh 2, matriks 3 x 3:
MASUKKAN ORDO MATRIKS A :
JUMI-AH BABIS MATRIKS A = ? 3JUMLAH KOLOM MATRIKS A = ? 3MASUKKAN ELEMEN MATRIKS A :
BARISlKOLOMl?2BARISlKOLOM2?OBARISlKOLOM3?1BARIS2KOLOMl?3BARIS2KOLOM2?1BARIS2KOLOM3?O
BARIS3KOLOMlTlBARISsKOLOM2?OBARIS3KOLOM3TlMATRIKS A ; MATRIKS SATUAN (IDENTITAS)
2.N 0.00 r.00 1.00 0.00 0.00
3.00 1.00 0.00 0.00 1.00 0.00
1.00 0.00 1.00 0.00 0.00 1.00
MATRIKS SATUAN ( IDENTITAS ) ; INVERS MATRIKS A
1.00 0.@ 0.00 1.00 0.00 -1.000.00 1.00 0.00 -3.00 1.00 3.00 1:
Sistem Persamaan Linear(SPL)
A . I : B, di mana: A = Matriks koefisien
X = Matriks variabel
B = Matriks suku tetap
Penyelesaian SPL salah satunya dengan cara OBE (operasi bariselementer):
OBE(AlB) -----+ (llx)
CONTOH
1. Tentukan SPL di bawah ini!
2Xr-3)h=-4
3Xr*5X2-13
OBE(AlB)-----+ (lX)
XB
EAB
E
t-3 -31 t#l = [#]
-115
.r b21( 3)-z.t13 j
-56
t-:,-;l;,'l 31,"lt-l
57
ft -t)Lo si
01
823(1)
---..-.--.'
Program sistem persamaan linear diambil dari program invers NxNdengan sedikit mengalami modifikasi.Jika pada invers matrik dengan ordo NxN ada N baris dan 2N kolom,maka pada SPL ada N baris dan N+1 kolom.
58
USTING PROGRAIUI:
10 rPRrM CHB$(16)EO NEM PROGRAM SISTEM PEBSAIIAAN UNE,AN
60 cfft40 PBIM ''tr/[,ASI'KKA}I On,DO I\IAItsIKS A :''00 INPTIT "JITMLAH BAruS MATAJKS A : ";I60 INPII "JITMLAH K0IOM IIAIAIKS A = "il90 DIM X(r,J)130 PRIM ''MASI'KKAN EIAMEN I\I[ATX,IKS A :''l40r0RB:]t0 I1ts0FORK:1T0J160 PBINT "BABIS ";B;"K0[OM ";K;170 INruT X(B,r)180 NEXI rI9O PBIMzOO NEXT B
eIO LPruM "MATBIKS KOEflSIEN A ";TA8(16*I);''MATEIKS SI'trU TETAP''
AEO IPRIM8S0r0BB:1T0IA40r0BK:1T0qI860 IPB;IM x(B,K),860 NEXT K
?70 LPRIIIT : LPH,IM
280 NEXT B
a90 LPnnn : LPBIM600FOBM:lTOI0I0 Dx : x(M,M)A20FORK=1T0JU60 x(M,K) = X(M'K) / DX
540 NEXT K
660 I'0BB:1T0 I500IrB:MTHEN410070 Dx = x(B,M)880FOBK=1T0tI890 X(B,K) : X(B,K) - DX * X(M,K)4OO NEXT K
410 NEXT B
480 MXr M.{50 LPBNT "tr[ATnilIG SATUA]I ( IDEIImA^S )";TAB(L6*I;;
''[i[,AIBJKS VABIABEL X''440 tPruM460FORB:1T0I
_21 b1201)
2J
-tl1l Xr =z)h =
2. Tentukan SPL di bawah ini!
1
2
r#{ tilXz=21 z 1
X3 -l I t -lx -3 L-1 z
A.X=B
!_122-1 I2-1
L_L22.1 .l-l; I;
*q
^l i I
-2 t2
r I +-+lo-tit+L -1 z'-l
0
-1-ti1]X,4lx24Jx3
=l-!,=!,
Ir olo IL6 z+
001001ri
2xr+x2-X1-X2*-X+2X-t23
OBE(AlB) ----------+
-il
(r-x)
I 2 1 -1I r -r 1L-1 z -l
1'2-1-11
21 b1(1/2) |. lll---------------+l IsJ L-r
1l b31(-1) r1oI---------------- lo3) Lo
1'l btzel)
2l --'
1)b21(-1)I l------------3J
1)b2(-2t3)0 ) ----------------
4)
1 \b32(-5t2)O ) -------------
-t)L2
1
2itsIr olo IL6 o il
0
-11
-35
I
400fOBK=I10J4?0 LPBIM X(B,r),480 }IEXT K490 LPBIM : LPBIM600 t{Exr B
610 I,PBIM : I,PBIM620FORB=110I6C0 LPBII{T "X";B;" = "r((8,.I)6{0 IJHM6O0 NEXT B
660 Et[D
HASIL PROGRAMDua persamaan dengan dua variabel:
MASUKK,AN ORDO MATR]KS A :
JUMLAH BARIS MATRIKS A : ? 2JUMLAH KOLOM MATRIKS A : ? 3MASUKKAN ELEMEN MATRIKS A :
BARISlKOLOMl?2BARIS 1 KOLOM 2? _3BARIS l KOLOM 3? _4
BARIS2KOLOMl?3BARIS2KOLOM2?5BARIS2KOLOM3?13
MATRIKS KOEFISIEN A
2
3
0
X1=1X2:2
HASIL PROGRAIII
Tiga persamaan dengan tiga variabel:
MASUKKAN ORDO MATRIKS A :
JUMLAH BARIS MATRIKS A : ? 3
60
JUMLAH KOLOM MATRIKS A: ? 4MASUKKAN ELEMEN MATRIKS A :
BARISlKOLOMl?1BARIS l KOLOM 2?.1BARISlKOLOM3?1BARISlKOLOM4?1BARIS2KOLOMl?2BARIS2KOLOM2?1BARIS2KOLOM3?_1BARIS2KOLOM4?2
BARIS3KOLOMl?_1BARIS3KOLOM2?2BARIS3KOLOMS?_1BARIS3KOLOM4?3
MATRIKS KOEFISIEN A
-11
2
MATRIKS SUKU TETAP
1
2
3
MATRIKS VARIABEL X
1
4
4
1
-1
-1
0
1
1
2
-1
MATRIKS SUKU TETAP
MATRIKS SATUAN ( IDENTITAS )
1000
0
X1:1X2 = 4X3=4
1
0
PROGRAIUI PERKALTAN MATRIKS DENGAN BII.ANGANSr(Ar-AR (SIGIIAT.BAS)
IO CIs8O REM PBOOMM PEN AIIAN MAIBITS A DENGAN BII,AIIGAN STAI,AB KC0 INPtn "il[ASlrrufA[ BILANCTAN SI(ALAB : ";X40 PBIMBO PBIM ''MASI'IffiAI,I OB,DO TIAIIIK9 A :''60 INPII ",II]MLAH BABIS = "8
-413
MATRIKS SATUAN ( IDENTITAS ) MATRIKS VARIABEL X
101
61
?0 INPIJT "tIIIMLAH K0t0M : "J(80 PRII{T
00r0BI:tT0B100FORJ:IltK110 PRINI "MBIS ";I;" KOIOM "pI;UIO INPIII A(I,qI)160 NEXI rI140 PBINI o
160 NEXT I160 PRIM : PRIM170 P&IM "BILANGAN SI(ALAB : ";XI8O PN,IM
190 PBNT "trf,ATRII(S A :"e0080RI=IT0B810FOR,I:1T0K2e0 PRIM USING "++ "$(L,I)iffiO NEXT J240 PRIM : PRIMPoO NEXT IP6O PRIM : PRIM
90 B(J,I) : A(I,,r)lOO NEXT JUO PN,IM
UIO MXT I140 PBIM : PBIM14I PBII{T "MATBIKS A : "I4er0BI:llOB146rORJ:110K144 PRIM USING "++ "A(I,,I);146 NEliSI eI
146 PBINT : PBIM14? NEXT I148 PBIM : PBIM160 PruIiE "IT,AIISP0SE MATBIKS A :"160 PRINT
I70rORI:1T0KI80 i'0n ,I : I I0 B
1S0 PBIM USING "++ ";B(I,J);EOO IIHIT .I810 PBIM : PBIMEEO NEXT I250 END
E?O PRIM ''PEN,I(ALLAN IfATA;IKS A DENEdN BIL. SI(AI,AB K : ''880FOBI:11O8e90FORtI:1TOK600B(I'.I) =XxA(I'.I)010 PRII{T USING "++ ";B(I,II);520 I{EXT ,,
560 PBINT : PBIM
040 I{EXT IA60 END
PROGRAM TRANSPOSE MATRTKS (TRANS.BAS)
10 cut20 NEM PBOGR,AM IBANSPOSE
21 PBIM ''MASIIKKA}I OB,DO MATN,UIS A :''ee INnn "IIIIMLAH BARIS MATruKS A : ";BeE INPUT "TIIIMLAH KOLOM MATRIKS A : ";K24 PBIM60 DrM A(B,r),8(K,B)4I PRIM ''il[A,9I'trKAIiI ETEMEN MAIBIK$ A :''50r0BI:IT0B60FOB,I:IT0K70 PRIlflt "MBIS "J;"K0[OM "iI;71 INPUT A(r,cr)
62 63