'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