![Page 1: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/1.jpg)
REVIEW OPERASI MATRIKS
T E K N I K L I N G K U N G A N I T B
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MENGHITUNG INVERS MATRIKS
1
0
0
1
22221221
21221121
22121211
21121111
acac
acac
acac
acac
10
01
2221
1211
2221
1211
aa
aa
cc
cc
![Page 3: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/3.jpg)
DETERMINAN
Hanya untuk square matrices
Jika determinan = 0 matriks singular, tidak punya invers
bcaddc
ba
dc
ba
det
123213312132231321
321
321
321
det
cbacbacbacbacbacba
ccc
bbb
aaa
![Page 4: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/4.jpg)
CARI INVERS NYA…
52
42
42
21
![Page 5: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/5.jpg)
S I M U L T A N E O U S L I N E A R E Q U A T I O N S
SISTEM PERSAMAAN LINEAR
![Page 6: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/6.jpg)
METODE PENYELESAIAN
• Metode grafik
• Eliminasi Gauss
• Metode Gauss – Jourdan
• Metode Gauss – Seidel
• LU decomposition
![Page 7: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/7.jpg)
METODE GRAFIK
2
4
11
21
2
1
x
x2
-2
Det{A} 0 A
is nonsingular
so invertible
Unique
solution
![Page 8: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/8.jpg)
SISTEM PERSAMAAN YANG TAK TERSELESAIKAN
5
4
42
21
2
1
x
x
No solution Det [A] = 0, but system is inconsistent
Then this system of
equations is not solvable
![Page 9: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/9.jpg)
SISTEM DENGAN SOLUSI TAK TERBATAS
Det{A} = 0 A is singular
infinite number of solutions
8
4
42
21
2
1
x
x
Consistent so solvable
![Page 10: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/10.jpg)
ILL-CONDITIONED SYSTEM OF EQUATIONS
A linear system
of equations is
said to be “ill-
conditioned” if
the coefficient
matrix tends to
be singular
![Page 11: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/11.jpg)
ILL-CONDITIONED SYSTEM OF EQUATIONS
• A small deviation in the entries of A matrix, causes a
large deviation in the solution.
47.1
3
99.048.0
21
2
1
x
x
47.1
3
99.049.0
21
2
1
x
x
1
1
2
1
x
x
0
3
2
1
x
x
![Page 12: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/12.jpg)
GAUSSIAN ELIMINATION
CXA
Merupakan salah satu teknik paling populer
dalam menyelesaikan sistem persamaan
linear dalam bentuk:
Terdiri dari dua step
1. Forward Elimination of Unknowns.
2. Back Substitution
![Page 13: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/13.jpg)
FORWARD ELIMINATION
112144
1864
1525
Tujuan Forward Elimination adalah untuk
membentuk matriks koefisien menjadi Upper
Triangular Matrix
7.000
56.18.40
1525
![Page 14: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/14.jpg)
FORWARD ELIMINATION
Persamaan linear
n persamaan dengan n variabel yang tak diketahui
11313212111 ... bxaxaxaxa nn
22323222121 ... bxaxaxaxa nn
nnnnnnn bxaxaxaxa ...332211
. .
. .
. .
![Page 15: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/15.jpg)
CONTOH
83125
12312
71352
21232
8325
1232
7352
2232
4321
4321
4321
4321
xxxx
xxxx
xxxx
xxxx
matriks input
![Page 16: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/16.jpg)
FORWARD ELIMINATION
83125
12312
71352
21232
32
162
190
13140
92120
12
112
31
'
14
'
4
'
13
'
3
'
12
'
2
1'
1
5
2
2
2
RRR
RRR
RRR
RR
32
162
190
13140
92120
12
112
31
41599
4500
1973002
912
110
12
112
31
'
14
'
4
'
23
'
3
2'
2
1
'
1
219
4
2
RRR
RRR
RR
RR
![Page 17: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/17.jpg)
FORWARD ELIMINATION
12572
12143000
319
37100
291
2110
12
112
31
'
34
'
4
3'
3
2
'
2
1
'
1
45
3
RRR
RR
RR
RR
41599
4500
1973002
912
110
12
112
31
12572
12143000
319
37100
291
2110
12
112
31
1435721000
319
37100
291
2110
12
112
31
121434'
4
3
'
3
2
'
2
1
'
1
RR
RR
RR
RR
![Page 18: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/18.jpg)
BACK SUBSTITUTION
4
143572
319
37
29
21
12
12
3
4
4
43
432
4321
x
x
xx
xxx
xxxx
![Page 19: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/19.jpg)
GAUSS - JOURDAN
39743
22342
15231
6150
8120
15231
'
13
'
3
'
12
'
2
1
'
1
3
2
RRR
RRR
RR
142700
42110
152101
'
23
'
3
2
'
2
'
21
'
1
5
2
3
RRR
RR
RRR
6150
8120
15231
142700
42110
152101
4100
2010
1001
273'
3
'
32
'
2
'
31
'
1
21
21
RR
RRR
RRR
![Page 20: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/20.jpg)
WARNING..
655
901.33099.26
7710
321
123
21
xxx
xxx
xx
Dua kemungkinan kesalahan -Pembagian dengan nol mungkin terjadi pada langkah
forward elimination. Misalkan:
- Kemungkinan error karena round-off (kesalahan pembulatan)
![Page 21: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/21.jpg)
CONTOH
Dari sistem persamaan linear
515
6099.23
0710
3
2
1
x
x
x
6
901.3
7
=
Akhir dari Forward Elimination
1500500
6001.00
0710
3
2
1
x
x
x
15004
001.6
7
=
6
901.3
7
515
6099.23
0710
15004
001.6
7
1500500
6001.00
0710
![Page 22: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/22.jpg)
KESALAHAN YANG MUNGKIN TERJADI
Back Substitution
99993.015005
150043 x
5.1 001.0
6001.6 32
xx
3500.010
077 32
1
xx
x
15004
001.6
7
1500500
6001.00
0710
3
2
1
x
x
x
![Page 23: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/23.jpg)
CONTOH KESALAHAN
Bandung-kan solusi exact dengan hasil perhitungan
99993.0
5.1
35.0
3
2
1
x
x
x
X calculated
1
1
0
3
2
1
x
x
x
X exact
![Page 24: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/24.jpg)
IMPROVEMENTS
Menambah jumlah angka penting
Mengurangi round-off error (kesalahan pembulatan)
Tidak menghindarkan pembagian dengan nol
Gaussian Elimination with Partial Pivoting
Menghindarkan pembagian dengan nol
Mengurangi round-off error
![Page 25: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/25.jpg)
PIVOTING
pka ,npk
Eliminasi Gauss dengan partial pivoting mengubah tata urutan
baris untuk bisa mengaplikasikan Eliminasi Gauss secara Normal
How?
Di awal sebelum langkah ke-k pada forward elimination, temukan angka
maksimum dari:
nkkkkk aaa .......,,........., ,1
Jika nilai maksimumnya Pada baris ke p,
Maka tukar baris p dan k.
![Page 26: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/26.jpg)
PARTIAL PIVOTING
What does it Mean?
Gaussian Elimination with Partial Pivoting ensures that
each step of Forward Elimination is performed with the
pivoting element |akk| having the largest absolute value.
Jadi,
Kita mengecek pada setiap langkah apakah angka
mutlak yang dipakai untuk forward elimination (pivoting
element) adalah selalu paling besar
![Page 27: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/27.jpg)
PARTIAL PIVOTING: EXAMPLE
655
901.36099.23
7710
321
321
21
xxx
xxx
xx
Consider the system of equations
In matrix form
515
6099.23
0710
3
2
1
x
x
x
6
901.3
7
=
Solve using Gaussian Elimination with Partial Pivoting using five
significant digits with chopping
![Page 28: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/28.jpg)
PARTIAL PIVOTING: EXAMPLE
Forward Elimination: Step 1
Examining the values of the first column
|10|, |-3|, and |5| or 10, 3, and 5
The largest absolute value is 10, which means, to follow the
rules of Partial Pivoting, we don’t need to switch the rows
6
901.3
7
515
6099.23
0710
3
2
1
x
x
x
5.2
001.6
7
55.20
6001.00
0710
3
2
1
x
x
x
Performing Forward Elimination
![Page 29: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/29.jpg)
PARTIAL PIVOTING: EXAMPLE
001.6
5.2
7
6001.00
55.20
0710
3
2
1
x
x
x
Forward Elimination: Step 2
Examining the values of the first column
|-0.001| and |2.5| or 0.0001 and 2.5
The largest absolute value is 2.5, so row 2 is switched with
row 3
5.2
001.6
7
55.20
6001.00
0710
3
2
1
x
x
x
Performing the row swap
![Page 30: REVIEW OPERASI MATRIKS - kuliah.ftsl.itb.ac.idkuliah.ftsl.itb.ac.id/wp-content/uploads/2016/10/Slide02S.pdf · Jika determinan = 0 matriks singular, ... •Eliminasi Gauss •Metode](https://reader031.vdocument.in/reader031/viewer/2022011809/5ccccbee88c993db288d2c8a/html5/thumbnails/30.jpg)
PARTIAL PIVOTING: EXAMPLE
Forward Elimination: Step 2
Performing the Forward Elimination results in:
002.6
5.2
7
002.600
55.20
0710
3
2
1
x
x
x
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PARTIAL PIVOTING: EXAMPLE
002.6
5.2
7
002.600
55.20
0710
3
2
1
x
x
x
Back Substitution
Solving the equations through back substitution
1002.6
002.63 x
15.2
55.2 22
xx
010
077 32
1
xx
x
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PARTIAL PIVOTING: EXAMPLE
1
1
0
3
2
1
x
x
x
X exact
1
1
0
3
2
1
x
x
x
X calculated
Compare the calculated and exact solution
The fact that they are equal is coincidence, but it does
illustrate the advantage of Partial Pivoting
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SUMMARY
-Forward Elimination
-Back Substitution
-Pitfalls
-Improvements
-Partial Pivoting