Menyelesaikan persamaan linier dengan pendekatan matriks. ELIMINASI GAUSS

Menyelesaikan persamaan linier dengan pendekatan matriks.

ELIMINASI GAUSS
Persamaan Umum Gauss
a11X1 + a12X2 + a13X3 … a1nXn = b1 E1
a21X1 + a22X2 + a23X3 … a2nXn = b2 E2
a31X1 + a23X2 + a33X3 … a3nXn = b3 E3
=
am1X1 + am2X2 + am3X3 … amnXn = bn En

Contoh soal : Matriks 3X3
Gunakan eliminasi Gauss untuk menyelesaikan persamaan berikut
4X1 + 5X2 + 0X3 = 4 E1
2X1 – 2X2 + 3X3 = 8 E2
2X1 + 1X2 + 5X3 = 12 E3
Tahap Pertama : Eliminasi Maju

Langkah pertama, adalah dengan mengeliminasi X1 dari persamaan E2 dengan syarat a11 0.
Rumus : E2' = E2 – m21 . E1
E3' = E3 – m31 . E1
m21 = a21a11
m31 = a31a11

a21' = a21 -a21a11 . a11=0 a22' = a22 -a21a11 . a12
= -2-24 (5)
= -92

a23' = a23 -a21a11 . a13 b2' = b2 -a21a11 . b1
= 3-24 (0) = 8-24 (4)
= 3 = 6

Langkah kedua eliminasi X1 dari E3 dengan syarat a11 0.

a31' = a31 -a31a11 . a11=0 a32' = a32 -a31a11 . a12
= 1-24 (5)
= -32

a33' = a33 -a31a11 . a13 b3' = b3 -a31a11 . b1
= 5-24 (0) = 12-24 (4)
= 5 = 10

Setelah mengeliminasi X1 pada persamaan E2 dan E3, maka persamaan linear tersebut menjadi :
4X1 + 5X2 + 0X3 = 4 E1
-92 X2 + 3X3 = 6 E2'
-32 X2 + 5X3 = 10 E3'
Langkah selanjutnya adalah mengeliminasi X2 pada persamaan E3, dengan cara :
a32'' =a32' -a32'a22' . a22'
= -32--32-92 . -92
= 0
a33'' =a33' -a32'a22' . a23' b3'' =b3' -a32'a22' . b2'
= 5--32-92 . 3 = 10--32-92 . 6
= 4 = 8

Kemudian didapatkan :
X3 = b3''a33''
= 8 4
X3 = 2
-92 X2 + 3X3 = 6

-92 X2 + 32 = 6

X2 = 0

4 X1 + 5X2 + 0X3 = 4

4 X1 + 50 = 4

X1 = 1

2X1 + 5X2 + 4X3 = 4 E1
2X1 + 7X2 + 3X3 = 8 E2
9X1 – 2X2 + 6X3 = 12 E3

Rumus : E2' = E2 – m21 . E1
E3' = E3 – m31 . E1
m21 = a21a11
m31 = a31a11

a21' = a21 -a21a11 . a11=0 a22' = a22 -a21a11 . a12
= 7-22 (5)
= 2

a23' = a23 -a21a11 . a13 b2' = b2 -a21a11 . b1
= 3-22 (4) = 8-22 (4)
= -1 = 4


a31' = a31 -a31a11 . a11=0 a32' = a32 -a31a11 . a12
= -2-92 (5)
= -492

a33' = a33 -a31a11 . a13 b3' = b3 -a31a11 . b1
= 6-92 (4) = 12-92 (4)
= -12 = -6

2X1 + 5X2 + 4X3 = 4 E1
2X2 - X3 = 4 E2'
-492 X2 - 12X3 = -6 E3'
a32'' =a32' -a32'a22' . a22'
= -492--4922 . 2
= 0
a33'' =a33' -a32'a22' . a23' b3'' =b3' -a32'a22' . b2'
=-12--4922 . -1 =-6--4922 . 4
=-974 =43

X3 = b3''a33''
= 43 -974
X3 = -17297
2X2 - 1X3 = 4

2X2 - 1-17297 = 4

X2 = 10897

2X1 + 5X2 + 4X3 = 4

2X1 + 510897 + 4-17297 = 4

X1 = 26897

4X1 + 5X2 + 1X3 = 4 E1
3X1 + 9X2 - 3X3 = 8 E2
5X1 - 1X2 + 5X3 = 12 E3

Rumus : E2' = E2 – m21 . E1
E3' = E3 – m31 . E1
m21 = a21a11
m31 = a31a11

a21' = a21 -a21a11 . a11=0 a22' = a22 -a21a11 . a12
= 9-34 (5)
= 214

a23' = a23 -a21a11 . a13 b2' = b2 -a21a11 . b1
= -3-34 (1) = 8-34 (4)
= -154 = 5


a31' = a31 -a31a11 . a11=0 a32' = a32 -a31a11 . a12
= -1-54 (5)
= -294

a33' = a33 -a31a11 . a13 b3' = b3 -a31a11 . b1
= 5-54 (1) = 12-54 (4)
= 154 = 7

4X1 + 5X2 + 1X3 = 4 E1
214 X2 - 154X3 = 5 E2'
- 294 X2 - 154 X3 = 7 E3'
a32'' =a32' -a32'a22' . a22'
= - 294 -- 294 214 . 214
= 0

a33'' =a33' -a32'a22' . a23' b3'' =b3' -a32'a22' . b2'
= 154-- 294 214 . - 154 = 7-- 294 214 . 5
= -107 = 29221
X3 = b3''a33''
= 29221 -107
X3 = -14615

214 X2 -154X3 = 5

214 X2 -154-14615 = 5

X2 = -6

4X1 + 5X2 + 1X3 = 4

4X1 + 5-6 + 1-14615 = 4

X1 = 65660

4X1 + 2X2 + 3X3 + 5X4 = 4 E1
6X1 + 4X2 + 7X3 + 2X4 = 8 E2
1X1 + 5X2 + 0X3 + 2X4 = 12 E3
3X1 + 8X2 + 4X3 + 3X4 = 16 E4

Langkah pertama, adalah dengan mengeliminasi X1 dengan syarat a11 0.
Rumus : E2' = E2 – m21 . E1 E3' = E3 – m31 . E1
E4' = E4 – m41 . E1
m21 = a21a11 m31 = a31a11
m41 = a41a11

Pada E2 :

a21' = a21 -a21a11 . a11=0 a22' = a22 -a21a11 . a12
= 4-64 (2)
= 1

a23' = a23 -a21a11 . a13 a24' = a24 -a21a11 . a14
= 7-64 (3) = 2-64 (5)
= 104 = -224

b2' = b2 -a21a11 . b1
= 8-64 (4)
= 2

Pada E3 :

a31' = a31 -a31a11 . a11=0 a32' = a32 -a31a11 . a12
= 5-14 (2)
= 92

a33' = a33 -a31a11 . a13 a34' = a34 -a31a11 . a14
= 0-14 (3) = 2-14 (5)
= -34 = 34

b3' = b3 -a31a11 . b1
= 12-14 (4)
= 11

Pada E4 :

a41' = a41 -a41a11 . a11=0 a42' = a42 -a41a11 . a12
= 8-34 (2)
= 132

a43' = a43 -a41a11 . a13 a44' = a44 -a41a11 . a14
= 4-34 (3) = 3-34 (5)
= 74 = -34

b4' = b4 -a31a11 . b1
= 16-34 (4)
= 13

Setelah mengeliminasi X1 pada E2, E3, dan E4 maka persamaan linear tersebut menjadi :
4X1 + 2X2 + 3X3 + 5X4 = 4 E1
1X2 + 104 X3 - 224 X4 = 2 E2'
92 X2 - 34 X3 + 34 X4 = 11 E3'
132 X2 + 74 X3 - 34 X4 = 13 E4'

Langkah selanjutnya adalah mengeliminasi X2 pada persamaan E3 dan E4, dengan cara :
Pada E3 :
a32'' =a32' -a32'a22' . a22' a33'' =a33' -a32'a22' . a23'
= 92-921 . 1 = -34-921 .104
= 0 = -12
a34'' =a34' -a32'a22' . a24' b3'' =b3' -a32'a22' . b2'
= 34-921 .-224 = 11-921 . 2
= 1024 = 2

Pada E4 :
a42'' =a42' -a42'a22' . a22' a43'' =a43' -a42'a22' . a23'
= 132-1321 . 1 = 74-1321 .104
= 0 = -584
a44'' =a44' -a42'a22' . a24' b4'' =b4' -a42'a22' . b2'
= -34-1321 .-224 = 13-1321 . 2
= 35 = 0

Setelah mengeliminasi X2 pada E3, dan E4 maka persamaan linear tersebut menjadi :
4X1 + 2X2 + 3X3 + 5X4 = 4 E1
1X2 + 104 X3 - 224 X4 = 2 E2'
- 12X3 + 1024 X4 = 2 E3''
- 584 X3 + 35X4 = 0 E4''

Pada E4 :
a43''' =a43'' -a43''a33'' . a33''
= -584--584-12 . -12
= 0
a44''' =a44'' -a43''a33'' . a34'' b4''' =b4'' -a43''a33'' . b3''
= 35--584-12 .1024 = 0--584-12 . 2
= 20148 = -2912

Setelah mengeliminasi X3 pada E4 maka persamaan linear tersebut menjadi :
4X1 + 2X2 + 3X3 + 5X4 = 4 E1
1X2 + 104 X3 - 224 X4 = 2 E2'
- 12X3 + 1024 X4 = 2 E3''
- 20148 X4 = - 2912 E4'''
X4 = b4'''a44'''
= -2912 20148
X4 = -116201
- 12X3 + 1024 X4 = 2

- 12X3 + 1024 -116201 = 2

X3 = -280201

1X2 + 104 X3 - 224 X4 = 2

1X2 + 104 -280201 - 224 -116201 = 2

X2 = 464201

4X1 + 2X2 + 3X3 + 5X4 = 4

4 X1 + 2 464201 + 3 -280201 + 5 -116201 = 4

X1 = 324201

2X1 + 3X2 + 4X3 + 3X4 - 4X5 = 4 E1
4X1 + 5X2 + 6X3 + 1X4 - 4X5 = 8 E2
2X1 + 3X2 + 1X3 + 3X4 + 2X5 = 12 E3
0X1 + 2X2 + 5X3 + 1X4 - 2X5 = 16 E4
2X1 + 3X2 + 6X3 + 3X4 + 7X5 = 20 E5

Langkah pertama, adalah dengan mengeliminasi X1 pada E2 sampai E5 dengan syarat a11 0.
Rumus : E2' = E2 – m21 . E1 E3' = E3 – m31 . E1
E4' = E4 – m41 . E1 E5' = E5 – m51 . E1
m21 = a21a11 m31 = a31a11
m41 = a41a11 m51 = a51a11

Pada E2 :

a21' = a21 -a21a11 . a11=0 a22' = a22 -a21a11 . a12
= 5-42 (3)
= -1

a23' = a23 -a21a11 . a13 a24' = a24 -a21a11 . a14
= 6-42 (4) = 1-42 (3)
= -2 = -5

a25' = a25 -a21a11 . a15 b2' = b2 -a21a11 . b1
= -4-42 (-4) = 8-42 (4)
= 4 = 0

Pada E3 :

a31' = a31 -a31a11 . a11=0 a32' = a32 -a31a11 . a12
= 3-22 (3)
= 0

a33' = a33 -a31a11 . a13 a34' = a34 -a31a11 . a14
= 1-22 (4) = 3-22 (3)
= -3 = 0

a35' = a35 -a31a11 . a15 b3' = b3 -a31a11 . b1
= 2-22 (-4) = 12-22 (4)
= 6 = 8

Pada E4 :

a41' = a41 -a41a11 . a11=0 a42' = a42 -a41a11 . a12
= 2-02 (3)
= 2

a43' = a43 -a41a11 . a13 a44'= a44 -a41a11 . a14
= 5-02 (4) = 1-02 (3)
= 5 = 1

a45' = a45 -a41a11 . a15 b4' = b4 -a41a11 . b1
= -2-02 (-4) = 16-02 (4)
= -2 = 16

Pada E5 :

a51' = a51 -a51a11 . a11=0 a52' = a52 -a51a11 . a12
= 3-22 (3)
= 0

a53' = a53 -a51a11 . a13 a54' = a54 -a51a11 . a14
= 6-22 (4) = 3-22 (3)
= 2 = 0

a55' = a55 -a51a11 . a15 b5' = b5 -a51a11 . b1
= 7-22 (-4) = 20-22 (4)
= 11 = 16

Setelah mengeliminasi X1 pada E2, E3, E4, dan E5 maka persamaan linear tersebut menjadi :
2X1 + 3X2 + 4X3 + 3X4 - 4X5 = 4 E1
-1X2 - 2X3 - 5X4 + 4X5 = 0 E2'
0X2 - 3X3 + 0X4 + 6X5 = 8 E3'
-1X2 + 2X3 + 1X4 - 2X5 = 16 E4'
0X2 + 2X3 + 0X4 + 11X5 = 16 E5'

Langkah selanjutnya adalah mengeliminasi X2 pada persamaan E3, E4, dan E5 dengan cara :
Pada E3 :
a32'' =a32' -a32'a22' . a22' a33'' =a33' -a32'a22' . a23'
= 0-0-1 . -1 = -3-0-1 .-2
= 0 = -3

a34'' =a34' -a32'a22' . a24' a35'' =a35' -a32'a22' . a25'
= 0-0-1 .-5 = 6-0-1 .4
= 0 = 6

b3'' =b3' -a32'a22' . b2'
= 8-0-1 . 0
= 8

Pada E4 :
a42'' =a42' -a42'a22' . a22' a43'' =a43' -a42'a22' . a23'
= 2-2-1 . -1 = 5-2-1 .-2
= 0 = 1
a44'' =a44' -a42'a22' . a24' a45'' =a45' -a42'a22' . a25'
= 1-2-1 .-5 = -2-2-1 . 4
= -9 = 6

b4'' =b4' -a42'a22' . b2'
= 16-2-1 . 0
= 16

Pada E5 :
a52'' =a52' -a52'a22' . a22' a53'' =a53' -a52'a22' . a23'
= 0-0-1 . -1 = 2-0-1 .-2
= 0 = 2
a54'' =a54' -a52'a22' . a24' a55'' =a55' -a52'a22' . a25'
= 0-0-1 .-5 = 11-0-1 . 4
= 0 = 11

b5'' =b5' -a52'a22' . b2'
= 16-0-1 . 0
= 16

2X1 + 3X2 + 4X3 + 3X4 - 4X5 = 4 E1
-1X2 - 2X3 - 5X4 + 4X5 = 0 E2'
- 3X3 + 0X4 + 6X5 = 8 E3''
1X3 - 9X4 - 6X5 = 16 E4''
2X3 + 0X4 + 11X5 = 16 E5''

Pada E4 :
a43''' =a43'' -a43''a33'' . a32'' a44''' =a44'' -a43''a33'' . a34''
= 1-1-3 . -3 = -9-1-3 .0
= 0 = -9
a45''' =a45'' -a43''a33'' . a35'' b4''' =b4'' -a43''a33''. b3''
= 6-1-3 .6 = 16-1-3 . 8
= 8 = 563

Pada E5 :
a53''' =a53'' -a53''a33'' . a32'' a54''' =a54'' -a53''a33'' . a34''
= 2-2-3 . -3 = 0-2-3 .0
= 0 = 0
a55''' =a55'' -a53''a33'' . a35'' b5''' =b5'' -a53''a33'' . b3''
= 11-2-3 .6 = 16-2-3 . 8
= 15 = 643

Setelah mengeliminasi X3 pada E4 dan E5 maka persamaan linear tersebut menjadi :
2X1 + 3X2 + 4X3 + 3X4 - 4X5 = 4 E1
-1X2 - 2X3 - 5X4 + 4X5 = 0 E2'
- 3X3 + 0X4 + 6X5 = 8 E3''
-9X4 + 8X5 = 563 E4'''
0X4 + 15X5 = 643 E5'''


a54'''' =a54''' -a54'''a44''' . a44'''
= 0-0-9 . -9
= 0
a55'''' =a55''' -a54'''a44''' . a45''' b5'''' =b5''' -a54'''a44''' . b4'''
= 15-0-9 .8 = 643 -0-9 .40
= 15 = 643
Setelah mengeliminasi X4 pada E5 maka persamaan linear tersebut menjadi :

2X1 + 3X2 + 4X3 + 3X4 - 4X5 = 4 E1
-1X2 - 2X3 - 5X4 + 4X5 = 0 E2'
- 3X3 + 0X4 + 6X5 = 8 E3''
-9X4 + 8X5 = 563 E4'''
15X5 = 643 E5''''
X5 = b5''''a55''''
= 643 15
X5 = 6445
-9X4 + 8X5 = 563

-9X4 + 86445 = 563

X4 = -328405

- 3X3 + 0X4 + 6X5 = 8

- 3X3 + 0-328405 + 66445 = 8

X3 = 845

-1X2 - 2X3 - 5X4 + 4X5 = 0

-1X2 - 2845 - 5-328405 + 46445 = 0

X2 = 3800405
2X1 + 3X2 + 4X3 + 3X4 - 4X5 = 4
2X1 + 33800405 + 4845 + 3-328405 - 46445 = 4
X1 = -3390405

4X1 + 8X2 + 4X3 + 4X4 + 0X5 + 8X6 = 4 E1
5X1 + 3X2 + 6X3 + 2X4 + 1X5 + 4X6 = 8 E2
3X1 + 2X2 + 1X3 + 1X4 + 4X5 + 5X6 = 12 E3
0X1 + 2X2 + 1X3 + 4X4 + 3X5 + 5X6 = 16 E4
2X1 + 1X2 + 4X3 + 3X4 + 5X5 + 6X6 = 20 E5
1X1 + 4X2 + 3X3 + 5X4 + 6X5 + 2X6 = 24 E6

Langkah pertama, adalah dengan mengeliminasi X1 pada E2 sampai E6 dengan syarat a11 0.
Rumus : E2' = E2 – m21 . E1 E3' = E3 – m31 . E1
E4' = E4 – m41 . E1 E5' = E5 – m51 . E1
E6' = E6 – m61 . E1
m21 = a21a11 m31 = a31a11
m41 = a41a11 m51 = a51a11
m61 = a61a11

Pada E2 :

a21' = a21 -a21a11 . a11=0 a22' = a22 -a21a11 . a12
= 3-54 (8)
= -7

a23' = a23 -a21a11 . a13 a24' = a24 -a21a11 . a14
= 6-54 (4) = 2-54 (4)
= 1 = -3

a25' = a25 -a21a11 . a15 a26' = a26 -a21a11 . a16
= 1-54 (0) = 4-54 (8)
= 1 = -6

b2' = b2 -a21a11 . b1
= 8-54 (4)
= 3

Pada E3 :

a31' = a31 -a31a11 . a11=0 a32' = a32 -a31a11 . a12
= 2-34 (8)
= -4

a33' = a23 -a31a11 . a13 a34' = a24 -a31a11 . a14
= 1-34 (4) = 1-34 (4)
= -2 = -2

a35' = a25 -a31a11 . a15 a36' = a26 -a31a11 . a16
= 4-34 (0) = 5-34 (8)
= 4 = -1

b3' = b3 -a31a11 . b1
= 12-34 (4)
= 9

Pada E4 :

a41' = a41 -a41a11 . a11=0 a42' = a42 -a41a11 . a12
= 2-04 (8)
= 2

a43' = a43 -a41a11 . a13 a44' = a44 -a41a11 . a14
= 1-04 (4) = 4-04 (4)
= 1 = 4

a45' = a45 -a41a11 . a15 a46' = a46 -a41a11 . a16
= 3-04 (0) = 5-04 (8)
= 3 = 5

b4' = b4 -a41a11 . b1
= 16-04 (4)
= 16

Pada E5 :

a51' = a51 -a51a11 . a11=0 a52' = a52 -a51a11 . a12
= 1-24 (8)
= -3

a53' = a53 -a51a11 . a13 a54' = a54 -a51a11 . a14
= 4-24 (4) = 3-24 (4)
= 2 = 1

a55' = a55 -a51a11 . a15 a56' = a56 -a51a11 . a16
= 5-24 (0) = 6-24 (8)
= 5 = 2

b5' = b5 -a51a11 . b1
= 20-24 (4)
= 18

Pada E6 :

a61' = a61 -a61a11 . a11=0 a62' = a62 -a61a11 . a12
= 4-14 (8)
= 2

a63' = a63 -a61a11 . a13 a64' = a64 -a61a11 . a14
= 3-14 (4) = 5-14 (4)
= 2 = 4

a65' = a65 -a61a11 . a15 a66' = a66 -a61a11 . a16
= 6-14 (0) = 2-14 (8)
= 6 = 0

b6' = b6 -a61a11 . b1
= 24-14 (4)
= 23

Setelah mengeliminasi X1 pada E2, E3, E4,E5, dan E6 maka persamaan linear tersebut menjadi :
4X1 + 8X2 + 4X3 + 4X4 + 0X5 + 8X6 = 4 E1
-7X2 + 1X3 - 3X4 + 1X5 - 6X6 = 3 E2'
-4X2 - 2X3 - 2X4 + 4X5 - 1X6 = 9 E3'
2X2 + 1X3 + 4X4 + 3X5 + 5X6 = 16 E4'
-3X2 + 2X3 + 1X4 + 5X5 + 2X6 = 18 E5'
2X2 + 2X3 + 4X4 + 6X5 + 0X6 = 23 E6'

Langkah selanjutnya adalah mengeliminasi X2 pada persamaan E3, E4,E5, dan E6 dengan cara :
Pada E3 :
a32'' =a32' -a32'a22' . a22' a33'' =a33' -a32'a22' . a23'
= -4--4-7 . -7 = -2--4-7 .1
= 0 = -187

a34'' =a34' -a32'a22' . a24' a35'' =a35' -a32'a22' . a25'
= -2--4-7 .-3 = 4--4-7 .1
= -27 = 247

a36'' =a36' -a32'a22' . a26' b3'' =b3' -a32'a22' . b2'
= -1--4-7 . -6 = 9--4-7 . 3
= 177 = 517

Pada E4 :
a42'' =a42' -a42'a22' . a22' a43'' =a43' -a42'a22' . a23'
= 2-2-7 . -7 = 1-2-7 .1
= 0 = 97
a44'' =a44' -a42'a22' . a24' a45'' =a45' -a42'a22' . a25'
= 4-2-7 .-3 = 3-2-7 . 1
= 227 = 237

a46'' =a46' -a42'a22' . a26' b4'' =b4' -a42'a22' . b2'
= 5-2-7 . -6 = 16-2-7 . 3
= 237 = 1187
Pada E5 :
a52'' =a52' -a52'a22' . a22' a53'' =a53' -a52'a22' . a23'
= -3--3-7 . -7 = 2--3-7 .1
= 0 = 117

a54'' =a54' -a52'a22' . a24' a55'' =a55' -a52'a22' . a25'
= 1--3-7 .-3 = 5--3-7 . 1
= 167 = 327

a56'' =a56' -a52'a22' . a26' b5'' =b5' -a52'a22' . b2'
= 2--3-7 . -6 = 18--3-7 . 3
= 327 = 1177

Pada E6 :
a62'' =a62' -a62'a22' . a22' a53'' =a63' -a62'a22' . a23'
= 2-2-7 . -7 = 2-2-7 .1
= 0 = 97
a64'' =a64' -a62'a22' . a24' a65'' =a65' -a62'a22' . a25'
= 4-2-7 .-3 = 6-2-7 . 1
= 227 = 447

a66'' =a66' -a62'a22' . a26' b6'' =b6' -a62'a22' . b2'
= 0-2-7 . -6 = 23-2-7 . 3
= -127 = 1677

Setelah mengeliminasi X2 pada E3, E4, E5, dan E6 maka persamaan linear tersebut menjadi :
4X1 + 8X2 + 4X3 + 4X4 + 0X5 + 8X6 = 4 E1
-7X2 + 1X3 - 3X4 + 1X5 - 6X6 = 3 E2'
- 187X3 - 27X4 + 247X5 + 177X6 = 517 E3''
97X3 + 227X4 + 237X5 + 237X6 = 1187 E4''
117X3 + 167X4 + 327X5 + 327X6 = 1177 E5''
97X3 + 227X4 + 447X5 - 127X6 = 1677 E6''

Langkah selanjutnya adalah mengeliminasi X3 pada persamaan E4, E5, dan E6 dengan cara :
Pada E4 :
a43''' =a43'' -a43''a33'' . a33'' a44''' =a44'' -a43''a33'' . a34''
= 97-97-187 . -187 = 227-97-187 .-27
= 0 = 3
a45''' =a45'' -a43''a33'' . a35'' a46''' =a46'' -a43''a33''. a36''
= 237-97-187 .247 = 237-97-187 .177
= 5 = 92
b4''' =b4'' -a43''a32'' . b3''
= 1187-97-187 . 517
= 412
Pada E5 :
a53''' =a53'' -a53''a33'' . a33'' a54''' =a54'' -a53''a33'' . a34''
= 117-117-187 . -187 = 167-117-187 .-27
= 0 = 199
a55''' =a55'' -a53''a33'' . a35'' a56''' =a56'' -a53''a33''. a36''
= 327-117-187 .247 = 327-117-187 .177
= 203 = 10918
b5''' =b5'' -a53''a33'' . b3''
= 1177-117-187 . 517
= 1276

Pada E6 :
a63''' =a63'' -a63''a33'' . a33'' a64''' =a64'' -a63''a33'' . a34''
= 97-97-187 . -187 = 227-97-187 .-27
= 0 = 3

a65''' =a65'' -a63''a33'' . a35'' a66''' =a66'' -a63''a33''. a36''
= 447-97-187 .247 = -127-97-187 .177
= 8 = -12
b6''' =b6'' -a63''a33'' . b3''
= 1677-97-187 . 517
= 552

4X1 + 8X2 + 4X3 + 4X4 + 0X5 + 8X6 = 4 E1
-7X2 + 1X3 - 3X4 + 1X5 - 6X6 = 3 E2'
- 187X3 - 27X4 + 247X5 + 177X6 = 517 E3''
3X4 + 5X5 +92 X6 = 412 E4'''
199X4 + 203X5 + 10918X6 = 1276 E5'''
3X4 + 8X5 - 12X6 = 552 E6'''

Pada E5 :
a54'''' =a54''' -a54'''a44''' . a44''' a55'''' =a55''' -a54'''a44''' . a45'''
= 199-1993 . 3 = 203-1993 . 5 = 0 = 8527
a56'''' =a56''' -a54'''a44''' . a46''' b5'''' =b5''' -a54'''a44''' . b4'''
= 10918-1993 .92 = 1276 -1993 .412
= 5218 = 18227

Pada E6 :
a64'''' =a64''' -a64'''a44''' . a44''' a55'''' =a65''' -a64'''a44''' . a45'''
= 3-33 . 3 = 8-33 . 5 = 0 = 3

a66'''' =a66''' -a64'''a44''' . a46''' b6'''' =b6''' -a64'''a44''' . b4'''
= -12-33 .92 = 552 -33 .412
= -5 = 7

Setelah mengeliminasi X4 pada E5 dan E6 maka persamaan linear tersebut menjadi :

4X1 + 8X2 + 4X3 + 4X4 + 0X5 + 8X6 = 4 E1
-7X2 + 1X3 - 3X4 + 1X5 - 6X6 = 3 E2'
- 187X3 - 27X4 + 247X5 + 177X6 = 517 E3''
3X4 + 5X5 +92 X6 = 412 E4'''
8527X5 + 5218X6 = 18227 E5''''
3X5 -5X6 = 7 E6''''

Pada E6 :
a65''''' =a65'''' -a65''''a55'''' . a55'''' a66''''' =a66'''' -a65''''a55'''' . a45''''
= 3-38527 . 8527 = -5-38527 . 5218 = 0 = -65985
b6''''' =b6'''' -a65''''a55'''' . b4''''
= 7-38527 .18227
= 4985

X6 = b6'''''a66'''''
= 4985 -65985
X6 = -49659
8527X5 + 5218X6 = 18227

8527X5 + 5218-49659 = 18227

X5 = 7862435586

3X4 + 5X5 +92 X6 = 412

3X4 + 57862435586 +92 -49659 = 412

X4 = 11610035586

- 187X3 - 27X4 + 247X5 + 177X6 =517

- 187X3 - 2711610035586 + 2477862435586 + 177-49659 =517

X3 = -1139435586

-7X2 + 1X3 - 3X4 + 1X5 - 6X6 = 3
-7X2 + 1-1139435586 - 311610035586 + 17862435586 - 6-49659 = 3
X2 = -5313635586

4X1 + 8X2 + 4X3 + 4X4 + 0X5 + 8X6 = 4
4X1 + 8-5313635586 + 4-1139435586 + 411610035586
+ 07862435586 + 8-49659 = 4

X1 = 4244435586

Menyelesaikan persamaan linier dengan pendekatan matriks. ELIMINASI GAUSS

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