Download - AL Modul 2 Invers Matrik
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MODUL 2INVERS MATRIK
PRAYUDI STT PLN
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PENGERTIAN INVERS MATRIK
Matrik bujur sangkar A dikatakan !"un#ai in$!rs% jika t!rda"at
atrik & s!d!ikian ru"a s!'ingga (
A& ) &A ) I
diana I atrik id!ntitas
& dikatakan in$!rs atrik A ditu*is A+,% aka% AA+, ) A+,A ) I
A dikatakan in$!rs atrik & ditu*is &+,% aka% &+,&) &&+,) I
-.nt.' / A& ) &A ) I
,,,
201,02
30
40202,
=
,,,
201,02
=
,11
1,111,
30
40202,
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TEKNIK MENGHITUNG INVERS
M!t.d! Adj.int atrik
M!t.d! ."!rasi !*!!nt!r baris
M!t.d! P!rka*ian In$!rs Matrik E*!!nt!r
M!t.d! "artisi atrik
Pr.gra K."ut!r + MAT-ADS% MATLA&
5S O6I-E E7-ELL
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M!t.d! Adj.int Matrik
Andaikan A atrik bujur sangkar b!r.rd. 8n9n:% -ij)8;,:i
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Kasus% n ) 0
=
00020,
20222,
,0,2,,
aaa
aaa
aaa
A
=
=
0020,0
0222,2
0,2,,,
0020,0
0222,2
0,2,,,,;
MM;M
M;MM;
MM;M
d!t8A:
,
---
---
---
d!t8A:
,A
CONTOH :
=
334340
402
A d!t8A:) ,
=
=
=
,;2,;
2;3
,;33;
@:;8,2:;8,1;,:;8,3
,2:;8,1;:,;,1821:;8,3;
,:;8,321:;8,3;23:;821
40
02
34
02;
34
40
30
42;
34
42
34
30;
34
40
33
40;
33
34
8,:,A ,;
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KASUS ( n ) 4
=
=
440424,4
400020,0
420222,2
4,0,2,,,
,;
4440424,
0400020,
2420222,
,4,0,2,,
MM;MM;
M;MM;MMM;MM;
M;MM;M
d!t8A:, A
aaaa
aaaaaaaa
aaaa
A
-ONTOB (
Bitung*a' in$!rs atrik b!rikut ini (
Eks"ansi baris ;, (
d!t8a:)M,,;2M,2
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INVERS : OPERASI ELEMENTER BARIS
O"!rasi E*!!nt!r baris #ang
digunakan ada*a' (
8,:? BjkBj
82:? BjBi
80:? BjBj < kBj
,111
???,??????
11,1
111,
???
?????????
???
???
2,
2222,
,,2,,
nnnn
ii
n
n
aaa
a
aaa
aaa
Langka';*angka' s!bagai b!rikut
8,:? &!ntuk atrik *!ngka" A%I
82:? D!ngan s!rangkain ."!rasi
!*!*!t!r baris r!duksi*a' A%I
!njadi atrik b!rb!ntuk I%&
80:? A1 ) &
nnnn
ii
n
n
bbb
b
bbb
bbb
???
?????????
???
???
,111
???,??????
11,1
111,
2,
2222,
,,2,,
nnnn
ii
n
n
bbb
b
bbb
bbb
???
?????????
???
???
AFadi%
2,
2222,
,,2,,
,;O"!rasi !*!!nt!r baris
Ga.uss;F.rdan
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CONTOH :
M?Asa*
2 0 4 , 1 1
0 4 3 1 , 1
4 3 3 1 1 ,
It!rasi;,
, ,?3 2 1?3 1 1 B,)8,a,,:B,
1 ;1?3 ;, ;,?3 , 1 B2)B2;8a2,a,,:B,1 ;, ;0 ;2 1 , B0)B0;8a0,a,,:B,
It!rasi;2
, ,?3 2 1?3 1 1
1 , 2 0 ;2 1 B2)8,a22:B21 1 ;, , ;2 , B0)B0;8a02a22:B2
It!rasi;0
, ,?3 2 1?3 1 1
1 , 2 0 ;2 1
1 1 , ;, 2 ;, B0)8,a00:B0
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It!rasi;4
, ,?3 1 2?3 ;4 2 B,)B,;8a,0a00:B01 , 1 3 ; 2 B2)B2;8a20a00:B0
1 1 , ;, 2 ;,
It!rasi;3
, 1 1 ;3 3 ;, B,)B,;8a,2a22:B2
1 , 1 3 ; 2
1 1 , ;, 2 ;,
Lanjutan (
= ,;2,;
2;3
,;33;
AFadi% ,;
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Matrik AHa*
, 2 0 4 , 1 1 1
2 0 3 3 1 , 1 1
0 3 C 4 1 1 , 10 1 1 1 ,It!rasi ; ,
, 2 0 4 , 1 1 1 B,)8,a,,:B,
1 ;, ;, ;0 ;2 , 1 1 B2)B2;8a2,a,,:B,
1 ;, ;2 ; ;0 1 , 1 B0)B0;8a2,a,,:B,1 1 ;, ; ;0 1 1 , B4)B4;8a4,a,,:B,
It!rasi ; 2
, 2 0 4 , 1 1 1
1 , , 0 2 ;, 1 1 B2)8,a22:B2
1 1 ;, ;3 ;, ;, , 1 B0)B0;8a02a22:B21 1 ;, ; ;0 1 1 , B4)B4;8a42a22:B2
It!rasi;4
, 2 0 4 , 1 1 1
1 , , 0 2 ;, 1 1
1 1 , 3 , , ;, 1 B0)8,a00:B01 1 1 ;, ;2 , ;, , B4)B4; a40a00 B0
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It!rasi;3
, 2 0 4 , 1 1 1
1 , , 0 2 ;, 1 1
1 1 , 3 , , ;, 1
1 1 1 , 2 ;, , ;, B4)8,a44:B4It!rasi;
, 2 0 1 ;C 4 ;4 4 B,)B,;a,4B4
1 , , 1 ;4 2 ;0 0 B2)B2;a24B4
1 1 , 1 ;@ ; 3 B0)B0;a04B4
1 1 1 , 2 ;, , ;,It!rasi;C
, 2 1 1 21 ;,4 ,4 ;,, B,)B,;a,0B0
1 , 1 1 3 ;4 0 ;2 B2)B2;a20B0
1 1 , 1 ;@ ; 3
1 1 1 , 2 ;, , ;,It!rasi;
, 1 1 1 ,1 ; ;C B,)B,;a,2B2
1 , 1 1 3 ;4 0 ;2
1 1 , 1 ;@ ; 3
1 1 1 , 2 ;, , ;,
=
,;,,2
3;@;
2;04;3
C;;,1
AFadi%
,;
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PERKALIAN MATRIK ELEMENTER
8,:? Matrik !*!!nt!r ada*a'
atrik #ang di"!r.*!' dari."!rasi !*!!nt!r #ang
dik!nakan "ada atrik
id!ntitas?
82:? S!tia" atrik !*!!nt!r
!"un#ai in$!rs% dan s!tia"
atrik bujur sangkar b!r.rd.
8n9n: #ang !"un#ai in$!rs
!ki$a*!n baris t!r'ada" atrik
id!ntitas I?
80:? Akibatn#a% jika (
EkEk1Ek2JE2E1A ) I%
aka%
A1) EkEk1Ek2JE2E1
Matrik !*!!nt!r E di"!r.*!' dari
trans=.rasi atrik id!ntitas diana
"ada k.*. k!;I diganti d!ngann.ra*itas $!kt.r k.*. (
=
,??????11
???????????????????
1??????11???????????????????
1??????,1
1??????1,
?
?
?
?
ik
ik
ik
ik
i
N
N
N
N
E
=
iiin
aii
iii
ik
aa
aa
N
???
,
???
%
%,
%
iiiik IAEEEN ,2,% ???
(diana
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CONTOHBitung in$!rs atrik A
Jawab :
M!ng'itung E,
=
334
340
402
A
=
=
=
3
3
4
A/
3
4
0
A/
4
0
2
A 02,
=
=
,12;
1,,?3;
111?3
,1aa;
1,aa;
11,a
E
,,0,,,2,
,,
,
M!ng'itung E2
=
=
=
=
=
=
,2;,
12;0
104;
,12
1,,?3;
111?3
,2;1
12;1
10,
EE
,2;112;1
10,
,:8;,:8;1?3;111?3;,1
11?3;,?3;,
E
,;1?3;
,?3
34
0
,12;1,,?3;
111?3
AEN
,2
2
2,2
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Meng!"#ng E$ %an In&e'( Ma"'!k
,;2,;2;3
,;33;
,2;,12;0
104;
,;112,1
,;1,
EEE
,;112,1
,;1,
,G8;,:1182:G8;,:;,1
8;,:G8;,:;1,
E
,;2
,;
33
4
,2;,12;0
104;
AEEN
,20
0
0,20
Fadi In$!rs Matrik
=
,;2,;
2;3
,;33;
A ,;
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CONTOHBitung in$!rs atrik A
Jawab :
M!ng'itung E,
=
0
4C30
3302
402,
A
=
=
,110;
1,10;
11,2;
111,
E
0
0
2
,
AN
,
,,
M!ng'itung E2
=
=
=
=
=
=
,110;
1,,;,;
11,;2
1120;
,110;
1,10;
11,2;
111,
,111
1,,;1
11,;1
112,
EE
,111
1,,;1
11,;1
112,
,118;,:;1
1,8;,:8;,:;1
11,;,1
1128;,:;,
E
1,;
,;
2
3
0
2
,110;1,10;
11,2;
111,
AEN
,2
2
2,2
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M!ng'itung E0
=
=
=
=
=
=
,,;,2;
1,;,,
1,2;,
1,,0;
,110;
1,,;,;
11,;2
1120;
,111
1,11
11,1
111,
EEE
,,;11
1,;11
1,,1
1,1,
,8;,:8;,:;11
1,8;,:11
1,8;,:;,1
1,8;,:;1,
E
,;
,;,
,
C3
0
,110;
1,,;,;11,;2
1120;
AEEN
,20
0
0,20
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M!ng'itung E4 dan In$!rs Matrik
=
=
=
=
=
=
=
,;,,;2
3;@;
2;04;3
C;;,1
,,;,2;
1,;,,
1,2;,
1,,4;
,;111
3,11
2;1,1
C;11,
EEEEA
,;111
3,112;1,1
C;11,
,8;,:111
38;,:;,118;2:8;,:;1,1
8;C:8;,:;11,
E
,;
3
2;
C;
4
3
4
,,;,2;
1,;,,
1,2;,
1,,0;
AEEEN
,204,;
0
4,204
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INVERS : PARTISI MATRIK )1*Partisi atrik A #ang b!r.rd. 89n:
ada*a' sub atrik;sub atrik #angdi"!r.*!' dari A d!ngan ara
!b!rikan batasan;batasan garis
'.ris.nta* diantara dua baris dan
atau !b!rikan batasan;batasan
garis $!rtika* diantara dua k.*.?
CONTOH
=
0
4C303302
402,
A
=
=
=
=
4C A
0
30A
3340 A
022,A
(ada*a'AatrikPartisi
222,
,2,,
=
0,334
,00403002
20440
04302
A
CONTOH
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INVERS : PARTISI MATRIK )2*Andaikan A atrik bujur sangkar
b!r.rd. 8n9n: #ang !"un#aiin$!rs% #aitu ( A1) &% dan
"artisin#a asing;asing ada*a' (
Kar!na% A&)&A)I aka
di"!r.*!' (
=
=
222,
,2,,
222,
,2,,
&&
&&&/
AA
AAA
=
=
I1
1I
AA
AA
&&
&&
I1
1I
&&
&&
AA
AA
222,
,2,,
222,
,2,,
222,
,2,,
222,
,2,,
Dari "!rka*ian atrik di"!r.*!' 'asi* (
8,:? A11&11< A12&21) I
82:? A11&12< A12&22) 1
80:? &21A11< &22A21) 1
84:? &21A12< &22A22) I
D!ngan asusi% A111 ada% dan
&22) L1ada
Maka ruus untuk !ng'itung in$!ratrikn#a ada*a' (
8,:? &12 ) +8A 111A12:L
1
82:? &21) + L18A21A11
1:
80:? &11 ) A111
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CONTOH :Kasus n)4? Bitung*a' in$!rs
atrik b!rikut ini
FaHab (
=
0
4C30
3302
402,
A
=
=
=
=
4C A
0
30A
33
40 A
02
2,A
(ada*a'AatrikPartisi
222,
,2,,
Meng!"#ng L
=
=
=
=
=
=
=
=
=
=
,;,
3;
,;8;,:;
8;3:;;
3;
,LFadi%
;,;
3;,;
,2@
@;
4C
0,
2;,
0
30;
4C
AAAAL
10
,,
,;2
20;
0
30AA
0,
2;,
33
40
,;2
20;AA
,;2
20;
,2;
2;0
:408
,A
,;
,2,;
,,2,22
,;
,,2,
,2,;
,,
,;
,,
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Meng!"#ng In&e'( Ma"'!k
=
=
=
=
=
=
=
=
=
4;3
;,0
0;0
;,0
,;2
20;
,2;
;@
0,
2;,
,;2
20;
:A8AL:AA8A&
,;2
@;
10
,,
,;,
3;;
:A8A;L&
2;0
C;
,;,
3;
0,
2;,;
L:A;8A&
,;,,2,
,;,2
,;,,
,;,,,,
,;,,2,
,;2,
,;,2
,;,,,2
=
=
,;,,2
3;@;
2;04;3
C;;,1
&&
&&A
222,
,2,,,;
CONTOH :
Bitung in$!rs atrik A b!rikut (
FaHab ( Partisi atrik A
=
0,334
,0040
3002
20440
04302
A
=
=
=
=
0,3
,00
30
A
34
40
02
A
204
043 A
40
02A
222,
,2,,
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Meng!"#ng L
2,;
,1
1,
2;0
04;
34
40
02
AA
3C
;C;;
204
043
2;0
04;AA
2;0
04;
20;
0;4
@
,A
,;,,2,
,2,;
,,
,;,,
=
=
=
=
=
,;,;,
,;2;1
,1,;
,:;812:8;,;1:;8,
2:8;,;4:;822:8;2;
1:;8,2:8;2;,:;81
,
,L
Fadi%
2,;2
,;1,;
2,;,
,20
204
043
;
0,3
,00
30
3C;C;;
34
40
02
;
0,3
,00
30
AAAAL
,;
,2,;
,,2,22
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Meng!"#ng In&e'( Ma"'!k
=
=
=
=
=
=
=
=
=
2C;3
00C;
23;2
010;
2;0
04;
0;2
4;,
22;
3C;C;;
2;004;
:A8AL:AA8A&
02;
4,;
2;2
2,;
,1
1,
,;,;,
,;2;1
,1,;
;
:A8A;L&
4,C2
3;21;2;
,;,;,,;2;1
,1,;
3C
;C;;
;
L:A;8A&
,;,,2,
,;,2
,;,,
,;,,,,
,;,,2,
,;2,
,;,2
,;,,,2
=
=
,;,;,02;
,;2;14,;
,1,;2;2
4,C22C;3
3;21;2;00C;
&&&&AFadi%
222,,2,,,;
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SOAL TUGAS IV
Bitung in$!rs atrik A b!rikut ini d!ngan ara (
=
,,,4,,2
22,
,,,
bbaabbaa
aabb
aabb
A
a? M!t.d! Adj.int
b? P!rka*ian atrik !*!!nt!r
? O"!rasi !*!!nt!r baris
d? M!t.d! "artisi atrik
=
aaabb
aaabb
aaabbbbbaa
bbbaa
A
2,
2,
220,2,2,,
,2,2
Bitung in$!rs atrik A b!rikut ini d!ngan 2 ara "artisi #ang b!rb!da(
=
,0,,2
0,,20
,,204402,
02,,2
aaabb
aaabb
aaabbbbbaa
bbbaa
A