Matrices Inverse

Matrix Inverse dengan Metode Eliminasi Gauss JordanContoh Soal: MATRIX 3 x 3

3 1 2 3 0 0 3 0 0 297 0 0 297 0 0 1 0 0

5 1 -3 5 -2 -19 5 -2 0 495 -198 0 0 -198 0 0 1 0

-4 1 3 -4 7 17 -4 7 -99 0 0 -99 0 0 -99 0 0 1

1 0 0 1 -1 -2 1 -1 15 39 6 15 54 6 15

0 1 0 0 3 0 0 3 -57 228 -102 -57 -27 -102 -57

0 0 1 0 0 3 0 0 6 -24 42 6 81 42 6 R1 3R2-R1 3R3-2R1 R1 R2 2R3-19R2 99R1-4R3 99R2-7R3 R3 R1-2.5R2 R2 R3 R1/297 R2/-198 R3/-99

0.182 -0.09 0.273 -0.03 0.515 -0.21 -0.15 0.576 -0.06

1

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1 -3 4 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0

Tugas: MATRIX 4 x 4 1 1 -2 1 -1 8 -5 -2 1 1 -5 -5 1 1 0 0 1 1 0 0 1 29 0 0 -696 -174 0 0 0 -174 0 0 0 1 0 0 1 4 4 -2 1 4 24 18 1 4 24 0 1 116 696 0 0 0 696 0 0 0 696 0 0 0 1 0

0 2 -5 1 0 2 -5 1 0 2 5 11 0 2 5 -29 0 0 0 -29 0 0 0 -29 0 0 0 -29 0 0 0 1

1 0 0 0 1 3 -4 0 1 3 11 15 1 3 11 -27 1 33 184 -27 -512 -14 184 -27 456 -14 184 -27

0 1 0 0 0 1 0 0 0 1 5 5 0 1 5 -5 0 19 120 -5 120 6 120 -5 -96 6 120 -5

0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 3 0 6 44 3 44 8 44 3 -12 8 44 3

0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 -4 0 -8 -20 -4 -20 28 -20 -4 132 28 -20 -4 R1 R2+3R1 R3+4R1 R4 R1 R2 R3+5R2 R4+5R2 R1 R2 R3 3R3-4R4 R1 2R4+29R2 5R4+29R3 R4 R3-696R1 R3-6R2 R3 R4 4R2-R1 R2 R3 R4 R1/696 R2/-174 R3/696 R4/-29

-696 0 0 0 696 0 0 0 1 0 0 0

0.6552 -0.138 -0.017 0.1897 0.0805 -0.034 -0.046 -0.161 0.2644 0.1724 0.0632 -0.029 0.931 0.1724 -0.103 0.1379

2

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MATRIX 5x51 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 3 0 -1 0 1 3 0 -1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 3 0 0 0 0 3 0 0 0 0 1 0 0 0 0 0 2 2 0 0 0 2 2 0 0 0 2 2 0 0 0 2 0 0 0 0 2 0 0 0 0 6 0 0 0 0 12 0 0 0 0 1 0 0 0 0 -3 -1 1 0 0 -3 -1 1 0 0 -3 -1 1 0 0 -3 2 1 0 0 -3 2 0 0 0 -9 6 0 0 0 0 6 0 0 0 0 1 0 0 -1 -2 0 2 0 -1 -2 0 2 0 -1 1 0 1 0 -1 1 -1 1 0 -1 1 -1 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 1 1 0 0 0 0 1 0 0 0 -1 1 0 0 0 -1 1 0 0 0 -1 1 0 0 0 -2 1 -2 2 -2 -2 1 -2 -2 -2 -2 1.000 -0.667 -0.167 -0.333 -0.667 0 1 0 0 0 0 1 0 0 0 0 1 -3 0 1 0 1 -3 3 1 0 1 -3 3 -1 0 2 -8 8 -1 0 2 8 8 -1 0.000 0.667 0.667 1.333 -0.333 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 -1 0 0 0 1 -1 1 0 1 2 -2 1 0 1 -2 -2 1 0.000 0.333 -0.167 -0.333 0.333 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 -1 0 -1 1 2 -1 0 -1 8 2 -1 0.000 -0.333 0.667 0.333 -0.333 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 2 0 2 -2 2 2 0 2 2 2 2 0.000 0.667 0.167 0.333 0.667 R1 R2 R3 R4 R5-R1 R1 R2 R3-3R2 R4 R5+R2 R1 R2 R3 R4-R3 R5 R1 R2 R3 R4 2R5-R4 R1 3R2+R5 3R3-R5 3R4+R5 R5 R1 R2 2R3+3R4 R4 R5 R1/1 R2/3 R3/12 R4/6 R5/3

3

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Matrix Inverse dengan Metode AdjoinSoal: matrix 4 x 4 1 -3 4 0 1 -2 -1 -5 1 1 8 -2 0 2 -5 1

A=

Solve: Cari cof ( A ): M11 = (-1)1+1

-2 -1 -5 1 -1 -5 1 -2 -5 1 -2 -1 -3 4 0 1 4 0 1 -3 0 1 -3 4

1 8 -2 1 8 -2 1 1 -2 1 1 8 1 8 -2 1 8 -2 1 1 -2 1 1 8

2 -5 1 0 -5 1 0 2 1 0 2 -5 2 -5 1 0 -5 1 0 2 1 0 2 -5

=

1

x

114

= 114

M21

=

(-1)

2+1

=

-1

x

24

=

-24

M31

=

(-1)

3+1

=

1

x

-3

=

-3

M41

=

(-1)

4+1

=

-1

x

-33

=

33

M12

=

(-1)

1+2

=

-1

x

-14

=

14

M22

=

(-1)

2+2

=

1

x

-6

=

-6

M32

=

(-1)

3+2

=

-1

x

8

=

-8

M42

=

(-1)

4+2

=

1

x

-28

=

-28

4

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M13

=

(-1)

1+3

-3 4 0 1 4 0 1 -3 0 1 -3 4 -3 4 0 1 4 0 1 -3 0 1 -3 4

-2 -1 -5 1 -1 -5 1 -2 -5 1 -2 -1 -2 -1 -5 1 -1 -5 1 -2 -5 1 -2 -1

2 -5 1 0 -5 1 0 2 1 0 2 -5 1 8 -2 1 8 -2 1 1 -2 1 1 8

=

1

x

46

=

46

M23

=

(-1)

2+3

=

-1

x

-30

=

30

M33

=

(-1)

3+3

=

1

x

11

=

11

M43

=

(-1)

4+3

=

-1

x

5

=

-5

M14

=

(-1)

1+4

=

-1

x

-162

= 162

M24

=

(-1)

2+4

=

1

x

30

=

30

M34

=

(-1)

3+4

=

-1

x

18

=

-18

M44

=

(-1)

4+4

=

1

x

24

=

24

5

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Cof ( A ) =

114 -24 -3 33

14 -6 -8 -28

46 30 11 -5

162 30 -18 24

Adj ( A ) Adj ( A )

= = 114 14 46 162

{ Cof ( A ) }T -24 -3 -6 -8 30 11 30 -18

33 -28 -5 24

Inv ( A )

=

1 / Det ( A ) x Adj ( A ) 114 14 46 162 0.655 0.080 0.264 0.931 -24 -6 30 30 -0.138 -0.034 0.172 0.172 -3 -8 11 -18 -0.017 -0.046 0.063 -0.103 33 -28 -5 24 0.190 -0.161 -0.029 0.138

=

1/174

x

Inv ( A )

=

6

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Soal: matrix 5

x51 0 0 0 1 0 2 2 0 0 2 2 0 0 0 2 0 0 0 2 0 0 0 2 2 0 -3 -1 1 0 -3 -1 1 0 0 -1 1 0 0 -3 1 0 0 -3 -1 0 1 3 0 -1 -1 -2 0 2 0 -2 0 2 0 -1 0 2 0 -1 -2 2 0 -1 -2 0 0 0 2 2 0 0 0 -3 -1 1 0 -1 -2 0 2

A=

Solve: Cari Cof ( A ) M11= (-1)1+1 1 3 0 -1 0 3 0 -1 0 1 0 -1 0 1 3 -1 0 1 3 0 = 1 x -4 = -4

M21=

(-1)1+1

= -1 x 0 = 0

M31=

(-1)1+1

=1x0=0

M41=

(-1)1+1

= -1 X 0 = 0

M51=

(-1)1+1

=1x0=0

7

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M12=

(-1)1+1

0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0

0 2 2 0 0 2 2 0 0 0 2 0 0 0 2 0 0 0 2 2

0 -3 -1 1 0 -3 -1 1 0 0 -1 1 0 0 -3 1 0 0 -3 -1

-1 -2 0 2 0 -2 0 2 0 -1 0 2 0 -1 -2 2 0 -1 -2 0

= -1 x -6 = 6

M22=

(-1)1+1

= 1 x -4 = -4

M32=

(-1)1+1

= -1 x 0 = 0

M42=

(-1)1+1

=1x0=0

M52=

(-1)1+1

= -1 x -6 = 6

8

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M13=

(-1)1+1

0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0

1 3 0 -1 0 3 0 -1 0 1 0 -1 0 1 3 -1 0 1 3 0

0 -3 -1 1 0 -3 -1 1 0 0 -1 1 0 0 -3 1 0 0 -3 -1

-1 -2 0 2 0 -2 0 2 0 -1 0 2 0 -1 -2 2 0 -1 -2 0

=1x0=0

M23=

(-1)1+1

= -1 x -4 = 4

M33=

(-1)1+1

= 1 x -1 = -1

M43=

(-1)1+1

= -1 x -3 = 3

M53=

(-1)1+1

=1x0=0

9

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M14=

(-1)1+1

0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0

1 3 0 -1 0 3 0 -1 0 1 0 -1 0 1 3 -1 0 1 3 0

0 2 2 0 0 2 2 0 0 0 2 0 0 0 2 0 0 0 2 2

-1 -2 0 2 0 -2 0 2 0 -1 0 2 0 -1 -2 2 0 -1 -2 0

= -1 x 0 = 0

M24=

(-1)1+1

=1x8=8

M34=

(-1)1+1

= -1 x 2 = -2

M44=

(-1)1+1

=1x2=2

M54=

(-1)1+1

= -1 x 0 = 0

10

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M15=

(-1)1+1

0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0

1 3 0 -1 0 3 0 -1 0 1 0 -1 0 1 3 -1 0 1 3 0

0 2 2 0 0 2 2 0 0 0 2 0 0 0 2 0 0 0 2 2

0 -3 -1 1 0 -3 -1 1 0 0 -1 1 0 0 -3 1 0 0 -3 -1

= 1 x -2 = -2

M25=

(-1)1+1

= -1 x 0 = 0

M35=

(-1)1+1

=1x2=2

M45=

(-1)1+1

= -1 x 2 = -2

M55=

(-1)1+1

= 1 x -2 = -2

11

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Cof ( A ) =

-4 0 0 0 0

6 -4 0 0 6

0 4 -1 3 0

0 8 -2 2 0

-2 0 2 -2 -2

Adj ( A ) = -4 6 0 0 -2 1 0 0 0 1 2

Adj ( A ) =

{ Cof ( A ) }T 0 0 0 -4 0 0 4 -1 3 8 -2 2 0 2 -2 0 1 3 0 -1 0 0 2 2 0 0 0 -3 -1 1

0 6 0 0 -2 0 -1 -2 0 2 1 0 0 0 1 0 1 3 0 -1 0 0 2 2 0 0 0 -3 -1 1

Det ( A ) =

Det ( A ) =

Inv ( A )

= 1 / Det ( A ) x Adj ( A ) -4 6 0 0 -2 -2 3 0 0 -1 0 -4 4 8 0 0 -2 2 4 0 0 0 -1 -2 2 0 0 -0.5 -1 1 0 0 3 2 -2 0 0 1.5 1 -1 0 6 0 0 -2 0 3 0 0 -1

=

1/2

x

Inv ( A )

=

12

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Result:Gauss Jordan Inverse Matrix 4x 4 Adjoint

0.6552 0.0805 0.2644 0.931

-0.138 -0.034 0.1724 0.1724

-0.017 -0.046 0.0632 -0.103

0.1897 -0.161 -0.029 0.1379

=

0.655 -0.138 -0.017 0.190 0.080 -0.034 -0.046 -0.161 0.264 0.172 0.063 -0.029 0.931 0.172 -0.103 0.138

Inverse Matrix 5x 5

1 -0.667 -0.167 -0.333 -0.667

0 0.667 0.667 1.333 -0.333

0 0.333 -0.167 -0.333 0.333

0 -0.333 0.667 0.333 -0.333

0 0.667 0.167 0.333 0.667

-2 3 0 0 -1

0 -2 2 4 0

0 0 -0.5 -1 1

0 0 1.5 1 -1

0 3 0 0-1

13

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