eliminasi gaus jordan

8/16/2019 Eliminasi Gaus Jordan

1/17

ELIMINASIGAUS JORDAN

Kelompok


2/17

ANGGOTA:1. Fahira Isi!a"a#$i %1&1'1(1

,. I Ge-e Respa ra#a"o/a Sa$i0a

%1&1'1(11+

). Te/ar A-i ra-a#a %1&1'1(


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PENDAHULUAN

Me#/er!aka# elimi#asi -e#/a# me#//#aka# $ras#2ormasi

eleme#$er 3aris 4 kolom -ari sa$ ma$riks5 "ai$ :• e#kara# 3aris 4 kolom ke6i -e#/a# 3aris 4 kolom ke6!

7i! %A+ a$a K i! %A+

• erkalia# 3aris 4 kolom ke6i -e#/a# skalar 8 7i%8+%A+

a$a K i%8+%A+

• e#am3aha# 3aris 4 kolom ke6i -e#/a# 8 kali 3aris 4

kolom ke6!

7i!%8+%A+ a$a K i!%8+%A+

U#$k me#9ari akar %solsi+ persamaa# li#ier siml$a#5me$o-e #merik "a#/ -i/#aka# a-alah :

• Elimi#asi Gass6Jor-a#

• I$erasi Gass6Sei-el

Te$api5 Metode Gauss-Jourdan -apa$ !/a -i/#aka##$k me#9ari i#0ers sa$ ma$riks.


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Metode Eliminasi Gauss Jorda

Me$o-e i#i merpaka# pe#/em3a#/a# me$o-eelimi#asi Gass5 ha#"a sa!a a/me#$e- ma$rik5pa-a se3elah kiri -i3ah me#!a-i ma$rik -ia/o#

e#"elesaia# -ari persamaa# li#ier siml$a# -ia$as a-ala#ilai -15-,5-)5;5-# -a# a$a:

nnnnnn

n

n

n

baaaa

baaaa

baaaa

baaaa

...

..................

...

...

...

321

33333231

22232221

11131211

nd

d

d

d

1...000

..................

0...100

0...010

0...001

3

2

1

nn d xd xd xd x ==== ,....,,,

332211


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E>AMLE )USING ELEMENTAR? RO@OERATIONS%,4'+

7,

7, 7)6)7

7)6)7,

0 113

9 2

2

17

2

7

=−

−=−

=++

z y

z y

z y x

27113

177 2

9 2

−=−

−=−

=++

z y

z y

z y x

−−

−−

271130

17720

9211

−−

−−

271130

10

9211

2

17

2

7


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E>AMLE )USING ELEMENTAR? RO@OERATIONS%)4'+

6, 7)

6, 7)

716 7,

716 7,

3

9 2

2

17

2

7

=

−=−

=++

z

z y

z y x

2

3

2

1

2

17

2

7

9 2

−=−

−=−

=++

z

z y

z y x

−−

−−

2

3

2

1

2

17

2

7

00

10

9211

−−

3100

10

9211

2

17

2

7


9/17

E>AMLE )USING ELEMENTAR? RO@OERATIONS%'4'+

Solsi B C 15 "C, -a# C)

7, *4, 7)

71 6 114,

7)

7, *4, 7)

71 6 114,

7)

3

2

1

=

=

=

z

y

x

3

217

27

235

211

=

−=−

=+

z

z y

z x

−−

3100

10

01

2

17

2

7

2

35

2

11

3100

2010

1001


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METODE GAUSS-JORDAN

DALAM MENCAR N!ERS SUATUMATR"S


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Sa$ ma$riks 3!r sa#/kar A 3eror-o %N > N+ -ika$amemiliki i#0ers ma$riks "a#/ 3eror-o %N > N+ !ika

3erlak hal6hal se3a/ai 3erik$ : - A # A-$ % N

- A-$ # A % N

Dima#a IN a-alah ma$riks i-e#$i$as "a#/ 3eror-o %N >

Disampi#/ i$ #$k -apa$ -ika$aka# sa$ ma$riksmemiliki i#0ers a-alah ma$riks $erse3$ harsmerpaka# ma$riks #o# si#/lar a$a -e$ermi#a# -ar

ma$riks $erse3$ $i-ak sama -e#/a# #ol.

Matri&s n'ers den(an Metode Gauss-Jourdan


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Dike$ahi sa$ ma$riks :

Te#$ka# i#0ers -ari ma$riks -ia$as

Jaa3 :

1. Mla6mla ma$riks $erse3$ -ira#/kai -e#/a# ma$ri-e#$i$as -e#/a# or-o "a#/ sama.

CONTOH SOAL

A=

[2 4 4

3 5 3

4 1 2 ]

A= [2 4 4 1 0 0

3 5 3 0 1 0

2 1 2 0 0 1 ]


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,. Te#$ka# pi0o$ -ari ma$riks A "ai$ A%151+ -e#/a#mem3a/i , #$k 3aris per$ama :

). Kra#/i 3aris ke-a -e#/a# ) kali 3aris per$ama3aris ke$i/a -e#/a# , kali 3aris per$ama.

CONTOH SOAL

A=[1 2 2 12 0 0

3 5 3 0 1 0

2 1 2 0 0 1 ]

A=

[1 2 2 12 0 0

0 −1 −3 −32 1 0

0 −3 −2 −1 0 1 ]


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'. Eleme# A%,5,+ sekara#/ me#!a-i pi0o$5 #$kme#olka# A%)5,+.U#$k i$ 3aris ke-a -ikalika# -e#/

H1 -a# 3aris ke$i/a -i$am3ah ) kali 3aris ke-a$erse3$.

&. Eleme# A%)5 )+ -i!a-ika# pi0o$5 -e#/a# mem3a/i *

#$k 3aris ke$i/a.

CONTOH SOAL

A=[1 2 2 12 0 0

0 1 3 32 −1 0

0 0 7 72 −3 1]

A= [1 2 2 12 0 0

0 1 3 32 −1 0

0 0 1 12 −37 17 ]


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. Eleme# A %)5)+ i#i kem-ia# me#olka# A%,5)+ -a#A%15)+

*. Terakhir A%,5,+ me#olka# A%15,+

CONTOH SOAL

A=

[1 2 0 −12 67 −27

0 1 0 0 27 −37

0 0 1 12 −37 17 ]

A=

[1 0 0 −12 27 47

0 1 0 0 27 −

37

0 0 1 12 −37 17 ]


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. Ja-i5 ma$riks i#0ers -ari ma$riks A a-alah

CONTOH SOAL

A−1=

[−12 27 47

0 27 −37

12 −37 17 ]

eliminasi gaus jordan

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