CONTOH SOAL INVERS MATRIK & SISTEM PERSAMAAN LINEAR
DAN PENYELESAIAN
Matematika Terapan 2
Invers Matrik dan Sistem
Persamaan Linear
1. Sistem Persamaan Linear Metode
Gauss-Jordan
x1 + x2
- 3x3 + 4x4 = 1
-x1 + x2 + 4x3 - 2x4
= 1
-x1 + x2 + 2x3 - x4
= 0
1
1 -3
4 1 1 1
-3 4 1 1 1
-3 4 1
-1 1 4
-2 1 b2= b1+b2 0
2 1 2
2 b2=1/2b2 0
1 1/2 1
1
-1 1 2
-1 0 b3= b1+b3 0
2 -1 3
1 0 2
-1 3 1
1 1
-3 4 1 1
1 -3 4
1
0 1 1/2 1
1 0
1 1/2 1
1
b3= -2b2+b3
c 0
0 -2 1
1 b3= -1/2b3 0 0
1 -1/2 1/2
x1 + x2 - 3x3 + 4x4
= 1
x1 + 3/4 -5/4
x4 - 3/2 - 3/2x4
+ 4x4 = 1
x1 - 3/4 + 5/4x4
= 1
x1 = 1 + 3/4
- 5/4x4
= 7/4 - 5/4x4
x4 = α → x1= 7/4
- 5/4
α
x2 + 1/2x3
+ x4 = 1
x2 + 1/2 (1/2
+ 1/2x4) + x4 = 1
x2 + 1/4 +1/4x4
+ x4 = 1
x2 + 1/4 +
5/4x4 = 1
x2= 1 - 1/4
- 5/4x4
=3/4 - 5/4x4
x4 = α → x2 = 3/4 - 5/4 α
x3 - 1/2x4
= 1/2
x3=1/2+1/2x4
x4=α→ x3=1/2 + 1/2
α
2. Sistem Persamaan Linear Metode Kombinasi
Linear
x1 - x2
= 4
-x1 + 4x2 = 6
-x1 + 4x2 = 6
1
-1 4 1
-1 4 1
-1 4 1
-1 4 b1=b2+b1
-1
4 6 b2= b1+b2 0 3
10 b2=1/3b2
0
1 10/3 0
1 10/3
-1
4 6 b3= b1+b3 0 3
10 0
3 10 b3=-3b2+b3 0 0
0
1 0 22/3 k1
0
1 10/3 k2
0
0 0
W
= k1.U + k2.V
1 -1
= 22/3
-1 + 10/3 4
-1 4
22/3 -10/3
= -22/3 +
40/3
-22/3 40/3
12/3 4
= 18/3 =
6
18/3 6
3. Sistem Persamaan Linear Metode Cramer
2X1-X2+2X3 =
4
X1+3X2-X3 =
2
3X1-X2-3X3 =
0
A =
Det
A =
= (18 + 3 – 2) – (18 + 2 + 3) = -40
Det A1 =
= (-36 + 0 – 4) – (0 + 4 +
6) = -50
Det
A2 =
=
(-12 -12 + 0) – (12 + 0 – 12) = -24
Det
A3 =
=
(0 – 6 – 4) – (36 – 4 + 0) = -42
X1 =
=
= 1,25
X2 =
=
= 0,6
X3 =
=
= 1,05
4. Invers Matrik Metode OBE
A =
A =
A =
A =
A =
A =
A =
A =
5. Invers Matrik Metode Adjoint
A =
Adj A =
=
=
=
Det
A =
= (42 + 16 + 16) – (28 + 4 – 96)
= 74 – (-64)
= 138
Invers A =
=
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