Matrix Inverse dengan Metode Eliminasi Gauss Jordan
Contoh Soal: MATRIX 3 x 3
3 5 -4 1 0 01 1 1 0 1 02 -3 3 0 0 1
3 5 -4 1 0 0 R10 -2 7 -1 3 0 3R2-R10 -19 17 -2 0 3 3R3-2R1
3 5 -4 1 0 0 R10 -2 7 -1 3 0 R20 0 -99 15 -57 6 2R3-19R2
297 495 0 39 228 -24 99R1-4R30 -198 0 6 -102 42 99R2-7R30 0 -99 15 -57 6 R3
297 0 0 54 -27 81 R1-2.5R20 -198 0 6 -102 42 R20 0 -99 15 -57 6 R3
1 0 0 0.182 -0.09 0.273 R1/2970 1 0 -0.03 0.515 -0.21 R2/-1980 0 1 -0.15 0.576 -0.06 R3/-99
Support: [email protected]
Tugas: MATRIX 4 x 41 1 1 0 1 0 0 0-3 -2 1 2 0 1 0 04 -1 8 -5 0 0 1 00 -5 -2 1 0 0 0 1
1 1 1 0 1 0 0 0 R10 1 4 2 3 1 0 0 R2+3R10 -5 4 -5 -4 0 1 0 R3+4R10 -5 -2 1 0 0 0 1 R4
1 1 1 0 1 0 0 0 R10 1 4 2 3 1 0 0 R20 0 24 5 11 5 1 0 R3+5R20 0 18 11 15 5 0 1 R4+5R2
1 1 1 0 1 0 0 0 R10 1 4 2 3 1 0 0 R20 0 24 5 11 5 1 0 R30 0 0 -29 -27 -5 3 -4 3R3-4R4
1 1 1 0 1 0 0 0 R10 29 116 0 33 19 6 -8 2R4+29R20 0 696 0 184 120 44 -20 5R4+29R30 0 0 -29 -27 -5 3 -4 R4
-696 -696 0 0 -512 120 44 -20 R3-696R10 -174 0 0 -14 6 8 28 R3-6R20 0 696 0 184 120 44 -20 R30 0 0 -29 -27 -5 3 -4 R4
696 0 0 0 456 -96 -12 132 4R2-R10 -174 0 0 -14 6 8 28 R20 0 696 0 184 120 44 -20 R30 0 0 -29 -27 -5 3 -4 R4
1 0 0 0 0.6552 -0.138 -0.017 0.1897 R1/6960 1 0 0 0.0805 -0.034 -0.046 -0.161 R2/-1740 0 1 0 0.2644 0.1724 0.0632 -0.029 R3/6960 0 0 1 0.931 0.1724 -0.103 0.1379 R4/-29
Support: [email protected]
MATRIX 5x51 0 0 0 0 1 0 0 0 00 1 0 0 -1 0 1 0 0 00 3 2 -3 -2 0 0 1 0 00 0 2 -1 0 0 0 0 1 01 -1 0 1 2 0 0 0 0 1
1 0 0 0 0 1 0 0 0 0 R10 1 0 0 -1 0 1 0 0 0 R20 3 2 -3 -2 0 0 1 0 0 R30 0 2 -1 0 0 0 0 1 0 R40 -1 0 1 2 -1 0 0 0 1 R5-R1
1 0 0 0 0 1 0 0 0 0 R10 1 0 0 -1 0 1 0 0 0 R20 0 2 -3 1 0 -3 1 0 0 R3-3R20 0 2 -1 0 0 0 0 1 0 R40 0 0 1 1 -1 1 0 0 1 R5+R2
1 0 0 0 0 1 0 0 0 0 R10 1 0 0 -1 0 1 0 0 0 R20 0 2 -3 1 0 -3 1 0 0 R30 0 0 2 -1 0 3 -1 1 0 R4-R30 0 0 1 1 -1 1 0 0 1 R5
1 0 0 0 0 1 0 0 0 0 R10 1 0 0 -1 0 1 0 0 0 R20 0 2 -3 1 0 -3 1 0 0 R30 0 0 2 -1 0 3 -1 1 0 R40 0 0 0 3 -2 -1 1 -1 2 2R5-R4
1 0 0 0 0 1 0 0 0 0 R10 3 0 0 0 -2 2 1 -1 2 3R2+R50 0 6 -9 0 2 -8 2 1 -2 3R3-R50 0 0 6 0 -2 8 -2 2 2 3R4+R50 0 0 0 3 -2 -1 1 -1 2 R5
1 0 0 0 0 1 0 0 0 0 R10 3 0 0 0 -2 2 1 -1 2 R20 0 12 0 0 -2 8 -2 8 2 2R3+3R40 0 0 6 0 -2 8 -2 2 2 R40 0 0 0 3 -2 -1 1 -1 2 R5
1 0 0 0 0 1.000 0.000 0.000 0.000 0.000 R1/10 1 0 0 0 -0.667 0.667 0.333 -0.333 0.667 R2/30 0 1 0 0 -0.167 0.667 -0.167 0.667 0.167 R3/120 0 0 1 0 -0.333 1.333 -0.333 0.333 0.333 R4/60 0 0 0 1 -0.667 -0.333 0.333 -0.333 0.667 R5/3
Support: [email protected]
Matrix Inverse dengan Metode AdjoinSoal: matrix 4 x 4
A =
1 1 1 0-3 -2 1 24 -1 8 -50 -5 -2 1
Solve:Cari cof ( A ):
-2 1 2M11 = (-1)1+1 -1 8 -5 = 1 x 114 = 114
-5 -2 1
1 1 0M21 = (-1)2+1 -1 8 -5 = -1 x 24 = -24
-5 -2 1
1 1 0M31 = (-1)3+1 -2 1 2 = 1 x -3 = -3
-5 -2 1
1 1 0M41 = (-1)4+1 -2 1 2 = -1 x -33 = 33
-1 8 -5
-3 1 2M12 = (-1)1+2 4 8 -5 = -1 x -14 = 14
0 -2 1
1 1 0M22 = (-1)2+2 4 8 -5 = 1 x -6 = -6
0 -2 1
1 1 0M32 = (-1)3+2 -3 1 2 = -1 x 8 = -8
0 -2 1
1 1 0M42 = (-1)4+2 -3 1 2 = 1 x -28 = -28
4 8 -5
Support: [email protected]
-3 -2 2M13 = (-1)1+3 4 -1 -5 = 1 x 46 = 46
0 -5 1
1 1 0M23 = (-1)2+3 4 -1 -5 = -1 x -30 = 30
0 -5 1
1 1 0M33 = (-1)3+3 -3 -2 2 = 1 x 11 = 11
0 -5 1
1 1 0M43 = (-1)4+3 -3 -2 2 = -1 x 5 = -5
4 -1 -5
-3 -2 1M14 = (-1)1+4 4 -1 8 = -1 x -162 = 162
0 -5 -2
1 1 1M24 = (-1)2+4 4 -1 8 = 1 x 30 = 30
0 -5 -2
1 1 1M34 = (-1)3+4 -3 -2 1 = -1 x 18 = -18
0 -5 -2
1 1 1M44 = (-1)4+4 -3 -2 1 = 1 x 24 = 24
4 -1 8
Support: [email protected]
Cof ( A ) =
114 14 46 162-24 -6 30 30-3 -8 11 -1833 -28 -5 24
Adj ( A ) = { Cof ( A ) }T
Adj ( A ) =
114 -24 -3 3314 -6 -8 -2846 30 11 -5162 30 -18 24
Inv ( A ) = 1 / Det ( A ) x Adj ( A )
= 1/174 x
114 -24 -3 3314 -6 -8 -2846 30 11 -5162 30 -18 24
Inv ( A ) =
0.655 -0.138 -0.017 0.1900.080 -0.034 -0.046 -0.1610.264 0.172 0.063 -0.0290.931 0.172 -0.103 0.138
Support: [email protected]
Soal: matrix 5 x 5
A =
1 0 0 0 00 1 0 0 -10 3 2 -3 -20 0 2 -1 01 -1 0 1 2
Solve: Cari Cof ( A )
M11= (-1)1+1
1 0 0 -1
= 1 x -4 = -43 2 -3 -20 2 -1 0-1 0 1 2
M21= (-1)1+1
0 0 0 0
= -1 x 0 = 03 2 -3 -20 2 -1 0-1 0 1 2
M31= (-1)1+1
0 0 0 0
= 1 x 0 = 01 0 0 -10 2 -1 0-1 0 1 2
M41= (-1)1+1
0 0 0 0
= -1 X 0 = 01 0 0 -13 2 -3 -2-1 0 1 2
M51= (-1)1+1
0 0 0 0
= 1 x 0 = 01 0 0 -13 2 -3 -20 2 -1 0
Support: [email protected]
M12= (-1)1+1
0 0 0 -1
= -1 x -6 = 60 2 -3 -20 2 -1 01 0 1 2
M22= (-1)1+1
1 0 0 0
= 1 x -4 = -40 2 -3 -20 2 -1 01 0 1 2
M32= (-1)1+1
1 0 0 0
= -1 x 0 = 00 0 0 -10 2 -1 01 0 1 2
M42= (-1)1+1
1 0 0 0
= 1 x 0 = 00 0 0 -10 2 -3 -21 0 1 2
M52= (-1)1+1
1 0 0 0
= -1 x -6 = 60 0 0 -10 2 -3 -20 2 -1 0
Support: [email protected]
M13= (-1)1+1
0 1 0 -1
= 1 x 0 = 00 3 -3 -20 0 -1 01 -1 1 2
M23= (-1)1+1
1 0 0 0
= -1 x -4 = 40 3 -3 -20 0 -1 01 -1 1 2
M33= (-1)1+1
1 0 0 0
= 1 x -1 = -10 1 0 -10 0 -1 01 -1 1 2
M43= (-1)1+1
1 0 0 0
= -1 x -3 = 30 1 0 -10 3 -3 -21 -1 1 2
M53= (-1)1+1
1 0 0 0
= 1 x 0 = 00 1 0 -10 3 -3 -20 0 -1 0
Support: [email protected]
M14= (-1)1+1
0 1 0 -1
= -1 x 0 = 00 3 2 -20 0 2 01 -1 0 2
M24= (-1)1+1
1 0 0 0
= 1 x 8 = 80 3 2 -20 0 2 01 -1 0 2
M34= (-1)1+1
1 0 0 0
= -1 x 2 = -20 1 0 -10 0 2 01 -1 0 2
M44= (-1)1+1
1 0 0 0
= 1 x 2 = 20 1 0 -10 3 2 -21 -1 0 2
M54= (-1)1+1
1 0 0 0
= -1 x 0 = 00 1 0 -10 3 2 -20 0 2 0
Support: [email protected]
M15= (-1)1+1
0 1 0 0
= 1 x -2 = -20 3 2 -30 0 2 -11 -1 0 1
M25= (-1)1+1
1 0 0 0
= -1 x 0 = 00 3 2 -30 0 2 -11 -1 0 1
M35= (-1)1+1
1 0 0 0
= 1 x 2 = 20 1 0 00 0 2 -11 -1 0 1
M45= (-1)1+1
1 0 0 0
= -1 x 2 = -20 1 0 00 3 2 -31 -1 0 1
M55= (-1)1+1
1 0 0 0
= 1 x -2 = -20 1 0 00 3 2 -30 0 2 -1
Support: [email protected]
-4 6 0 0 -20 -4 4 8 0
Cof ( A ) = 0 0 -1 -2 20 0 3 2 -20 6 0 0 -2
Adj ( A ) = { Cof ( A ) }T
-4 0 0 0 06 -4 0 0 6
Adj ( A ) = 0 4 -1 3 00 8 -2 2 0-2 0 2 -2 -2
Det ( A ) =
1 0 0 0 0 1 0 0 00 1 0 0 -1 0 1 0 00 3 2 -3 -2 0 3 2 -30 0 2 -1 0 0 0 2 -11 -1 0 1 2 1 -1 0 1
Det ( A ) = 2
Inv ( A ) = 1 / Det ( A ) x Adj ( A )
= 1/2 x
-4 0 0 0 06 -4 0 0 60 4 -1 3 00 8 -2 2 0-2 0 2 -2 -2
Inv ( A ) =
-2 0 0 0 03 -2 0 0 30 2 -0.5 1.5 00 4 -1 1 0-1 0 1 -1 -1
Support: [email protected]
Result:
Gauss Jordan AdjointInverse Matrix 4x 4 0.6552 -0.138 -0.017 0.1897
0.0805 -0.034 -0.046 -0.1610.2644 0.1724 0.0632 -0.0290.931 0.1724 -0.103 0.1379 =
0.655 -0.138 -0.017 0.1900.080 -0.034 -0.046 -0.1610.264 0.172 0.063 -0.0290.931 0.172 -0.103 0.138
Inverse Matrix 5x 5 1 0 0 0 0
-0.667 0.667 0.333 -0.333 0.667-0.167 0.667 -0.167 0.667 0.167-0.333 1.333 -0.333 0.333 0.333-0.667 -0.333 0.333 -0.333 0.667
≠
-2 0 0 0 03 -2 0 0 30 2 -0.5 1.5 00 4 -1 1 0-1 0 1 -1 -1
Support: [email protected]
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