Persamaan Cremeeer

8/12/2019 Persamaan Cremeeer

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PERSAMAAN CREMER

PERSAMAAN MATRIK:

NO.1 :

X1 X2 X3 X4 b

1 2 3 1 2 11

2 3 4 2 6 12

3 2 5 1 2 13

4 4 2 2 2 -14

SOLUTION:

MATRIX 1 ( A)

Sign A1 A2 A3 A4

+

1 2 3 1 2

2 3 4 2 6

3 2 5 1 2

4 4 2 2 2


2/44

Eliminate elements in the 1st column under the 1st element

Sign A1 A2 A3 A4

+

1 2 3 1 2

2 0 -0.5 0.5 3

3 0 2 0 0

4 0 -4 0 -2

Eliminate elements in the 2nd column under the 2nd element

Sign A1 A2 A3 A4

+

1 2 3 1 2

2 0 -0.5 0.5 3

3 0 0 2 12

4 0 0 -4 -26

Eliminate elements in the 3rd column under the 3rd element

Sign A1 A2 A3 A4

+

1 2 3 1 2

2 0 -0.5 0.5 3

3 0 0 2 12


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4 0 0 0 -2

Multiply the main diagonal elements

Sign A1 A2 A3 A4

+

1 2 3 1 2

2 0 -0.5 0.5 3

3 0 0 2 12

4 0 0 0 -2

2 x (-0.5) x 2 x (-2) = 4

Result:

Determinant is 4

MATRIX 2 ( B)

Sign A1 A2 A3 A4

+

1 11 3 1 2

2 12 4 2 6

3 13 5 1 2

4 -14 2 2 2


4/44


Sign A1 A2 A3 A4

+

1 11 3 1 2

2 0 0.7272727272727275 0.9090909090909092 3.8181818181818183

3 0 1.454545454545455 -0.18181818181818188 -0.36363636363636376

4 0 5.818181818181818 3.2727272727272725 4.545454545454545


Sign A1 A2 A3 A4

+

1 11 3 1 2

2 0 0.7272727272727275 0.9090909090909092 3.8181818181818183

3 0 0 -2.0000000000000004 -8

4 0 0 -3.999999999999999 -25.999999999999993


Sign A1 A2 A3 A4

+

1 11 3 1 2

2 0 0.7272727272727275 0.9090909090909092 3.8181818181818183


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3 0 0 -2.0000000000000004 -8

4 0 0 0 -10


Sign A1 A2 A3 A4

+

1 11 3 1 2

2 0 0.7272727272727275 0.9090909090909092 3.8181818181818183

3 0 0 -2.0000000000000004 -8

4 0 0 0 -10

11 x 0.7272727272727275 x (-2.0000000000000004) x (-10) = 160.0000000000001

Result:

Determinant is 160.0000000000001

MATRIX 3 ( C)

Sign A1 A2 A3 A4

+

1 2 11 1 2

2 3 12 2 6

3 2 13 1 2

4 4 -14 2 2


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Sign A1 A2 A3 A4

+

1 2 11 1 2

2 0 -4.5 0.5 3

3 0 2 0 0

4 0 -36 0 -2


Sign A1 A2 A3 A4

+

1 2 11 1 2

2 0 -4.5 0.5 3

3 0 0 0.2222222222222222 1.3333333333333333

4 0 0 -4 -26


Sign A1 A2 A3 A4

+

1 2 11 1 2

2 0 -4.5 0.5 3


7/44

3 0 0 0.2222222222222222 1.3333333333333333

4 0 0 0 -2


Sign A1 A2 A3 A4

+

1 2 11 1 2

2 0 -4.5 0.5 3

3 0 0 0.2222222222222222 1.3333333333333333

4 0 0 0 -2

2 x (-4.5) x 0.2222222222222222 x (-2) = 4

Result:

Determinant is 4

MATRIX 4 ( D)

Sign A1 A2 A3 A4

+

1 2 3 11 2

2 3 4 12 6

3 2 5 13 2

4 4 2 -14 2


8/44


Sign A1 A2 A3 A4

+

1 2 3 11 2

2 0 -0.5 -4.5 3

3 0 2 2 0

4 0 -4 -36 -2


Sign A1 A2 A3 A4

+

1 2 3 11 2

2 0 -0.5 -4.5 3

3 0 0 -16 12

4 0 0 0 -26


Sign A1 A2 A3 A4

+

1 2 3 11 2

2 0 -0.5 -4.5 3

3 0 0 -16 12


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4 0 0 0 -26

2 x (-0.5) x (-16) x (-26) = -416

Result:

Determinant is -416

MATRIX 5 ( E)

Sign A1 A2 A3 A4

+

1 2 3 1 11

2 3 4 2 12

3 2 5 1 13

4 4 2 2 -14


Sign A1 A2 A3 A4

+

1 2 3 1 11

2 0 -0.5 0.5 -4.5

3 0 2 0 2

4 0 -4 0 -36



10/44

Sign A1 A2 A3 A4

+

1 2 3 1 11

2 0 -0.5 0.5 -4.5

3 0 0 2 -16

4 0 0 -4 0


Sign A1 A2 A3 A4

+

1 2 3 1 11

2 0 -0.5 0.5 -4.5

3 0 0 2 -16

4 0 0 0 -32


Sign A1 A2 A3 A4

+

1 2 3 1 11

2 0 -0.5 0.5 -4.5

3 0 0 2 -16

4 0 0 0 -32

2 x (-0.5) x 2 x (-32) = 64


11/44

Result:

Determinant is 64

JAWABAN :

X1 = B / D = 160.0000000000001 / 4 = 40.00000000000003

X2 = C / D = 4 / 4 = 1

X3 = D / D = -416 / 4 = -104

X4 = D4 / D = 64 / 4 = 16

SOLUTION SET:

X1 = 40.00000000000003

X2 = 1

X3 = -104

X4 = 16

NO.2 :

X1 X2 X3 X4 X5 b

1 1 2 3 4 5 14

2 1 2 5 1 2 15

3 2 5 1 2 1 16

4 4 2 2 6 3 17

5 2 4 1 6 1 18


12/44

SOLUTION:

MATRIX 1 ( A)

Sign A1 A2 A3 A4 A5

+

1 1 2 3 4 5

2 1 2 5 1 2

3 2 5 1 2 1

4 4 2 2 6 3

5 2 4 1 6 1


Sign A1 A2 A3 A4 A5

+

1 1 2 3 4 5

2 0 0 2 -3 -3

3 0 1 -5 -6 -9

4 0 -6 -10 -10 -17

5 0 0 -5 -2 -9

Swap the 2nd and the 3rd rows inversing determinant sign

Sign A1 A2 A3 A4 A5


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-

1 1 2 3 4 5

2 0 1 -5 -6 -9

3 0 0 2 -3 -3

4 0 -6 -10 -10 -17

5 0 0 -5 -2 -9


Sign A1 A2 A3 A4 A5

-

1 1 2 3 4 5

2 0 1 -5 -6 -9

3 0 0 2 -3 -3

4 0 0 -40 -46 -71

5 0 0 -5 -2 -9


Sign A1 A2 A3 A4 A5

-

1 1 2 3 4 5

2 0 1 -5 -6 -9


14/44

3 0 0 2 -3 -3

4 0 0 0 -106 -131

5 0 0 0 -9.5 -16.5

Eliminate elements in the 4th column under the 4th element

Sign A1 A2 A3 A4 A5

-

1 1 2 3 4 5

2 0 1 -5 -6 -9

3 0 0 2 -3 -3

4 0 0 0 -106 -131

5 0 0 0 0 -4.759433962264151


Sign A1 A2 A3 A4 A5

-

1 1 2 3 4 5

2 0 1 -5 -6 -9

3 0 0 2 -3 -3

4 0 0 0 -106 -131

5 0 0 0 0 -4.759433962264151


15/44

-(1 x 1 x 2 x (-106) x (-4.759433962264151)) = -1008.9999999999999

Result:

Determinant is -1008.9999999999999

MATRIX 2 ( B)

Sign A1 A2 A3 A4 A5

+

1 14 2 3 4 5

2 15 2 5 1 2

3 16 5 1 2 1

4 17 2 2 6 3

5 18 4 1 6 1


Sig

n A1 A2 A3 A4 A5

+

1 14 2 3 4 5

2 0

-

0.14285714285

71428

1.7857142857142

86

-

3.2857142857

142856

-

3.35714285714

28577


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3 02.71428571428

57144

-

2.4285714285714

284

-

2.5714285714

28571

-

4.71428571428

5714

4 0

-

0.42857142857

14284

-

1.6428571428571

428

1.1428571428

571432

-

3.07142857142

8571

5 01.42857142857

14288

-

2.8571428571428

568

0.8571428571

428577

-

5.42857142857

1429


Sig

n

A

1A2 A3 A4 A5

+

11

42 3 4 5

2 0

-

0.142857142857

1428

1.785714285714

286

-

3.285714285714

2856

-

3.357142857142

8577

3 0 031.50000000000

0025

-

65.00000000000

003

-

68.50000000000

004

4 0 0 -7 117.000000000000

002

5 0 0 15.00000000000 - -


17/44


18/44

Sig

n

A

1A2 A3 A4 A5

+

11

42 3 4 5

2 0

-

0.142857142857

1428

1.785714285714

286

-

3.285714285714

2856

-

3.357142857142

8577

3 0 0

31.50000000000

0025

-

65.00000000000003

-

68.50000000000004

4 0 0 0

-

3.444444444444

4393

-

8.222222222222

22

5 0 0 0 0

-

3.8801843317972398


Sig

n

A

1A2 A3 A4 A5

+

11

42 3 4 5

2 0-

0.142857142857

1.785714285714

286

-

3.285714285714

-

3.357142857142


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1428 2856 8577

3 0 031.50000000000

0025

-

65.00000000000

003

-

68.50000000000

004

4 0 0 0

-

3.444444444444

4393

-

8.222222222222

22

5 0 0 0 0

-

3.8801843317972398

14 x (-0.1428571428571428) x 31.500000000000025 x (-3.4444444444444393) x (-

3.8801843317972398) = -842

Result:

Determinant is -842

Sign A1 A2 A3 A4 A5

+

1 1 14 3 4 5

2 1 15 5 1 2

3 2 16 1 2 1

4 4 17 2 6 3

5 2 18 1 6 1


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Sign A1 A2 A3 A4 A5

+

1 1 14 3 4 5

2 0 1 2 -3 -3

3 0 -12 -5 -6 -9

4 0 -39 -10 -10 -17

5 0 -10 -5 -2 -9


Sign A1 A2 A3 A4 A5

+

1 1 14 3 4 5

2 0 1 2 -3 -3

3 0 0 19 -42 -45

4 0 0 68 -127 -134

5 0 0 15 -32 -39


Sign A1 A2 A3 A4 A5


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+

1 1 14 3 4 5

2 0 1 2 -3 -3

3 0 0 19 -42 -45

4 0 0 0 23.31578947368422 27.05263157894734

5 0 0 0 1.1578947368421098 -3.473684210526315


Sign A1 A2 A3 A4 A5

+

1 1 14 3 4 5

2 0 1 2 -3 -3

3 0 0 19 -42 -45

4 0 0 0 23.31578947368422 27.05263157894734

5 0 0 0 0 -4.817155756207677


Sign A1 A2 A3 A4 A5

+

1 1 14 3 4 5

2 0 1 2 -3 -3

3 0 0 19 -42 -45


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4 0 0 0 23.31578947368422 27.05263157894734

5 0 0 0 0 -4.817155756207677

1 x 1 x 19 x 23.31578947368422 x (-4.817155756207677) = -2134.000000000002

MATRIX 3 ( C)

Sign A1 A2 A3 A4 A5

+

1 1 14 3 4 5

2 1 15 5 1 2

3 2 16 1 2 1

4 4 17 2 6 3

5 2 18 1 6 1


Sign A1 A2 A3 A4 A5

+

1 1 14 3 4 5

2 0 1 2 -3 -3

3 0 -12 -5 -6 -9

4 0 -39 -10 -10 -17


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5 0 -10 -5 -2 -9


Sign A1 A2 A3 A4 A5

+

1 1 14 3 4 5

2 0 1 2 -3 -3

3 0 0 19 -42 -45

4 0 0 68 -127 -134

5 0 0 15 -32 -39


Sign A1 A2 A3 A4 A5

+

1 1 14 3 4 5

2 0 1 2 -3 -3

3 0 0 19 -42 -45

4 0 0 0 23.31578947368422 27.05263157894734

5 0 0 0 1.1578947368421098 -3.473684210526315



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Sign A1 A2 A3 A4 A5

+

1 1 14 3 4 5

2 0 1 2 -3 -3

3 0 0 19 -42 -45

4 0 0 0 23.31578947368422 27.05263157894734

5 0 0 0 0 -4.817155756207677


Sign A1 A2 A3 A4 A5

+

1 1 14 3 4 5

2 0 1 2 -3 -3

3 0 0 19 -42 -45

4 0 0 0 23.31578947368422 27.05263157894734

5 0 0 0 0 -4.817155756207677

1 x 1 x 19 x 23.31578947368422 x (-4.817155756207677) = -2134.000000000002

Result:

Determinant is -2134.000000000002

MATRIX 4 ( D)


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Sign A1 A2 A3 A4 A5

+

1 1 2 14 4 5

2 1 2 15 1 2

3 2 5 16 2 1

4 4 2 17 6 3

5 2 4 18 6 1


Sign A1 A2 A3 A4 A5

+

1 1 2 14 4 5

2 0 0 1 -3 -3

3 0 1 -12 -6 -9

4 0 -6 -39 -10 -17

5 0 0 -10 -2 -9


Sign A1 A2 A3 A4 A5

-

1 1 2 14 4 5

2 0 1 -12 -6 -9


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3 0 0 1 -3 -3

4 0 -6 -39 -10 -17

5 0 0 -10 -2 -9


Sign A1 A2 A3 A4 A5

-

1 1 2 14 4 5

2 0 1 -12 -6 -9

3 0 0 1 -3 -3

4 0 0 -111 -46 -71

5 0 0 -10 -2 -9


Sign A1 A2 A3 A4 A5

-

1 1 2 14 4 5

2 0 1 -12 -6 -9

3 0 0 1 -3 -3

4 0 0 0 -379 -404

5 0 0 0 -32 -39


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Sign A1 A2 A3 A4 A5

-

1 1 2 14 4 5

2 0 1 -12 -6 -9

3 0 0 1 -3 -3

4 0 0 0 -379 -404

5 0 0 0 0 -4.889182058047496


Sign A1 A2 A3 A4 A5

-

1 1 2 14 4 5

2 0 1 -12 -6 -9

3 0 0 1 -3 -3

4 0 0 0 -379 -404

5 0 0 0 0 -4.889182058047496

-(1 x 1 x 1 x (-379) x (-4.889182058047496)) = -1853.000000000001

Result:

Determinant is -1853.000000000001

MATRIX 5 ( E)

Your matrix


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Sign A1 A2 A3 A4 A5

+

1 1 2 3 14 5

2 1 2 5 15 2

3 2 5 1 16 1

4 4 2 2 17 3

5 2 4 1 18 1


Sign A1 A2 A3 A4 A5

+

1 1 2 3 14 5

2 0 0 2 1 -3

3 0 1 -5 -12 -9

4 0 -6 -10 -39 -17

5 0 0 -5 -10 -9


Sign A1 A2 A3 A4 A5

-

1 1 2 3 14 5

2 0 1 -5 -12 -9


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3 0 0 2 1 -3

4 0 -6 -10 -39 -17

5 0 0 -5 -10 -9


Sign A1 A2 A3 A4 A5

-

1 1 2 3 14 5

2 0 1 -5 -12 -9

3 0 0 2 1 -3

4 0 0 -40 -111 -71

5 0 0 -5 -10 -9


Sign A1 A2 A3 A4 A5

-

1 1 2 3 14 5

2 0 1 -5 -12 -9

3 0 0 2 1 -3

4 0 0 0 -91 -131

5 0 0 0 -7.5 -16.5


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Sign A1 A2 A3 A4 A5

-

1 1 2 3 14 5

2 0 1 -5 -12 -9

3 0 0 2 1 -3

4 0 0 0 -91 -131

5 0 0 0 0 -5.703296703296703


Sign A1 A2 A3 A4 A5

-

1 1 2 3 14 5

2 0 1 -5 -12 -9

3 0 0 2 1 -3

4 0 0 0 -91 -131

5 0 0 0 0 -5.703296703296703

-(1 x 1 x 2 x (-91) x (-5.703296703296703)) = -1037.9999999999998

Result:

Determinant is -1037.9999999999998

MATRIX 6 ( E)


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Sign A1 A2 A3 A4 A5

+

1 1 2 3 4 14

2 1 2 5 1 15

3 2 5 1 2 16

4 4 2 2 6 17

5 2 4 1 6 18


Sign A1 A2 A3 A4 A5

+

1 1 2 3 4 14

2 0 0 2 -3 1

3 0 1 -5 -6 -12

4 0 -6 -10 -10 -39

5 0 0 -5 -2 -10


Sign A1 A2 A3 A4 A5


32/44

-

1 1 2 3 4 14

2 0 1 -5 -6 -12

3 0 0 2 -3 1

4 0 -6 -10 -10 -39

5 0 0 -5 -2 -10


Sign A1 A2 A3 A4 A5

-

1 1 2 3 4 14

2 0 1 -5 -6 -12

3 0 0 2 -3 1

4 0 0 -40 -46 -111

5 0 0 -5 -2 -10


Sign A1 A2 A3 A4 A5

-

1 1 2 3 4 14

2 0 1 -5 -6 -12

3 0 0 2 -3 1


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4 0 0 0 -106 -91

5 0 0 0 -9.5 -7.5


Sign A1 A2 A3 A4 A5

-

1 1 2 3 4 14

2 0 1 -5 -6 -12

3 0 0 2 -3 1

4 0 0 0 -106 -91

5 0 0 0 0 0.6556603773584904


Sign A1 A2 A3 A4 A5

-

1 1 2 3 4 14

2 0 1 -5 -6 -12

3 0 0 2 -3 1

4 0 0 0 -106 -91

5 0 0 0 0 0.6556603773584904

-(1 x 1 x 2 x (-106) x 0.6556603773584904) = 138.99999999999994

Result:


34/44

Determinant is 138.99999999999994

x1 = D1 / D = -842 / (-1008.9999999999999) = 0.8344895936570863

x2 = D2 / D = -2134.000000000002 / (-1008.9999999999999) = 2.1149653121902894

x3 = D3 / D = -1853.000000000001 / (-1008.9999999999999) = 1.8364717542120923

x4 = D4 / D = -1037.9999999999998 / (-1008.9999999999999) = 1.0287413280475717

x5 = D5 / D = 138.99999999999994 / (-1008.9999999999999) = -0.13776015857284435

Solution set:

x1 = 0.8344895936570863

x2 = 2.1149653121902894

x3 = 1.8364717542120923

x4 = 1.0287413280475717

x5 = -0.13776015857284435

PERSAMAAN GAUSS-JORDAN

ORDE 4 X 4

X1 X2 X3 X4 b

1 1 2 3 4 2

2 4 5 2 2 3


35/44

3 2 7 4 5 4

4 3 2 5 2 5

5 4 2 4 4 6

Find the pivot in the 1st column in the 1st row

X1 X2 X3 X4 b

1 1 2 3 4 2

2 4 5 2 2 3

3 2 7 4 5 4

4 3 2 5 2 5

5 4 2 4 4 6

Eliminate the 1st column

X1 X2 X3 X4 b

1 1 2 3 4 2

2 0 -3 -10 -14 -5

3 0 3 -2 -3 0


36/44

4 0 -4 -4 -10 -1

5 0 -6 -8 -12 -2

Make the pivot in the 2nd column by dividing the 2nd row by -3

X1 X2 X3 X4 b

1 1 2 3 4 2

2 0 1 10/3 14/3 5/3

3 0 3 -2 -3 0

4 0 -4 -4 -10 -1

5 0 -6 -8 -12 -2

Eliminate the 2nd column

X1 X2 X3 X4 b

1 1 0 -11/3 -16/3 -4/3

2 0 1 10/3 14/3 5/3

3 0 0 -12 -17 -5

4 0 0 28/3 26/3 17/3

5 0 0 12 16 8

Make the pivot in the 3rd column by dividing the 3rd row by -12


37/44

X1 X2 X3 X4 b

1 1 0 -11/3 -16/3 -4/3

2 0 1 10/3 14/3 5/3

3 0 0 1 17/12 5/12

4 0 0 28/3 26/3 17/3

5 0 0 12 16 8

Eliminate the 3rd column

X1 X2 X3 X4 b

1 1 0 0 -5/36 7/36

2 0 1 0 -1/18 5/18

3 0 0 1 17/12 5/12

4 0 0 0 -41/9 16/9

5 0 0 0 -1 3

Find the pivot in the 4th column (inversing the sign in the whole row) and swap the 5th and the

4th rows

X1 X2 X3 X4 b

1 1 0 0 -5/36 7/36


38/44

2 0 1 0 -1/18 5/18

3 0 0 1 17/12 5/12

4 0 0 0 1 -3

5 0 0 0 -41/9 16/9

Eliminate the 4th column

X1 X2 X3 X4 b

1 1 0 0 0 -2/9

2 0 1 0 0 1/9

3 0 0 1 0 14/3

4 0 0 0 1 -3

5 0 0 0 0 -107/9

A= -2/9

B= 1/9

C= 14/3

D= -3

E= -107/9

ORDE 5 X 5

X1 X2 X3 X4 X5 b


39/44

1 2 3 2 2 2 2

2 3 3 2 7 2 5

3 4 2 3 2 4 3

4 5 3 4 2 4 5

5 6 4 2 3 2 6

6 7 2 3 3 2 2

Make the pivot in the 1st column by dividing the 1st row by 2

X1 X2 X3 X4 X5 b

1 1 3/2 1 1 1 1

2 3 3 2 7 2 5

3 4 2 3 2 4 3

4 5 3 4 2 4 5

5 6 4 2 3 2 6

6 7 2 3 3 2 2

Eliminate the 1st column

X1 X2 X3 X4 X5 b

1 1 3/2 1 1 1 1


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2 0 -3/2 -1 4 -1 2

3 0 -4 -1 -2 0 -1

4 0 -9/2 -1 -3 -1 0

5 0 -5 -4 -3 -4 0

6 0 -17/2 -4 -4 -5 -5

Make the pivot in the 2nd column by dividing the 2nd row by -3/2

X1 X2 X3 X4 X5 b

1 1 3/2 1 1 1 1

2 0 1 2/3 -8/3 2/3 -4/3

3 0 -4 -1 -2 0 -1

4 0 -9/2 -1 -3 -1 0

5 0 -5 -4 -3 -4 0

6 0 -17/2 -4 -4 -5 -5

Eliminate the 2nd column

X1 X2 X3 X4 X5 b

1 1 0 0 5 0 3

2 0 1 2/3 -8/3 2/3 -4/3


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3 0 0 5/3 -38/3 8/3 -19/3

4 0 0 2 -15 2 -6

5 0 0 -2/3 -49/3 -2/3 -20/3

6 0 0 5/3 -80/3 2/3 -49/3

Make the pivot in the 3rd column by dividing the 3rd row by 5/3

X1 X2 X3 X4 X5 b

1 1 0 0 5 0 3

2 0 1 2/3 -8/3 2/3 -4/3

3 0 0 1 -38/5 8/5 -19/5

4 0 0 2 -15 2 -6

5 0 0 -2/3 -49/3 -2/3 -20/3

6 0 0 5/3 -80/3 2/3 -49/3

Eliminate the 3rd column

X1 X2 X3 X4 X5 b

1 1 0 0 5 0 3

2 0 1 0 12/5 -2/5 6/5

3 0 0 1 -38/5 8/5 -19/5


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4 0 0 0 1/5 -6/5 8/5

5 0 0 0 -107/5 2/5 -46/5

6 0 0 0 -14 -2 -10

Make the pivot in the 4th column by dividing the 4th row by 1/5

X1 X2 X3 X4 X5 b

1 1 0 0 5 0 3

2 0 1 0 12/5 -2/5 6/5

3 0 0 1 -38/5 8/5 -19/5

4 0 0 0 1 -6 8

5 0 0 0 -107/5 2/5 -46/5

6 0 0 0 -14 -2 -10

Eliminate the 4th column

X1 X2 X3 X4 X5 b

1 1 0 0 0 30 -37

2 0 1 0 0 14 -18

3 0 0 1 0 -44 57

4 0 0 0 1 -6 8


43/44


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6 0 0 0 0 0 -219/32

Dari operasi matematika diatas diperoleh nilai:

A= -9/32

B= 21/16

C= 13/32

D= -81/64

E= -219/32

Persamaan Cremeeer

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