eliminasi gaus jordan

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    ELIMINASI GAUS JORDAN

    Kelompok

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    ANGGOTA: 1. Fahira Isi!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#$er 3aris 4 kolom -ari sa$ 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#$k me#9ari akar %solsi+ persamaa# li#ier siml$a#5 me$o-e #merik "a#/ -i/#aka# a-alah :

    • Elimi#asi Gass6Jor-a#

    • I$erasi Gass6Sei-el

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

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    Metode Eliminasi Gauss Jorda

     Me$o-e i#i merpaka# pe#/em3a#/a# me$o-e elimi#asi Gass5 ha#"a sa!a a/me#$e- ma$rik5 pa-a se3elah kiri -i3ah me#!a-i ma$rik -ia/o#

     e#"elesaia# -ari persamaa# li#ier siml$a# -ia$as a-ala #ilai -15-,5-)5;5-# -a# a$a:

         

         

    nnnnnn

    n

    n

    n

    baaaa

    baaaa

    baaaa

    baaaa

    ...

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

    ...

    ...

    ...

    321

    33333231

    22232221

    11131211

      

     

      

     

    nd 

    1...000

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

    0...100

    0...010

    0...001

    3

    2

    1

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

    332211

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    E>AMLE ) USING ELEMENTAR? RO@ OERATIONS%,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>AMLE ) USING ELEMENTAR? RO@ OERATIONS%)4'+

    6, 7)

    6, 7)

    716 7,

    716 7,

    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

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    E>AMLE ) USING ELEMENTAR? RO@ OERATIONS%'4'+

      Solsi B C 15 "C, -a# C)

    7,  *4, 7)

    71 6 114,

    7)

    7,  *4, 7)

    71 6 114,

    7)

    =

    =

    =

     z 

     y

     x

    2 17

    2 7

    2 35

    2 11

    =

    −=−

    =+

     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 SUATU MATR"S

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     Sa$ ma$riks 3!r sa#/kar A 3eror-o %N > N+ -ika$a memiliki i#0ers ma$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$aka# sa$ ma$riks memiliki i#0ers a-alah ma$riks $erse3$ hars merpaka# 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$ahi sa$ ma$riks :

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

      Jaa3 :

     1. Mla6mla ma$riks $erse3$ -ira#/kai -e#/a# ma$r i-e#$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 :

     ). Kra#/i 3aris ke-a -e#/a# ) kali 3aris per$ama 3aris ke$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$5 #$k me#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 ]

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