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 - 5/4α
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 + 1/2α
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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