Pemuda Pencari Bakat

1.SISTEM PERSAMAAN LINIER DENGAN METODE ELIMINASI GAUSS
-2X1 2X2 1X3 3X4 = 17
3X1 -5X2 2X3 2X4 = 7
4X1 2X2 3X3 2X4 = 24
1X1 -3X2 2X3 -1X4 = -3
1.Metode Eliminasi Gauss
-2 2 1 3 17 b1x -1/2
3 -5 2 2 7
4 2 3 2 24
1 -3 2 -1 -3

1 -1 -1/2 -3/2 -17/2
3 -5 2 2 7 b2 -3.b1
4 2 3 2 24 b3 -4.b1
1 -3 2 -1 -3 b4 -b1
1 -1 -1/2 -3/2 -17/2
0 9 1/2 3/2 68/2 b2 x (1/9)
0 2 1/2 3 -170
0 -2 5/2 -10/2 - 280/51
1 -1 -1/2 -3/2 -17/2
0 1 1/18 3/8 34/9
0 2 1/2 3 -170 b3 – 4 .b2
0 -2 5/2 -10/2 - 280/51 b4 -17/2. b2

1 -1 -1/2 -3/2 -17/2
0 1 1/18 3/8 34/9
0 0 3/18 3/8 -5848/9 b3x 18/3
0 0 7/60 26/5 333217/630

1 -1 -1/2 -3/2 -17/2
0 1 1/18 3/8 34/9
0 0 1 9/4 -23392/3
0 0 7/60 26/5 630/280 b4-7/60 .b3

1 -1 -1/2 -3/2 -17/2 b1+3/2 . b4
0 1 1/18 3/8 34/9 b2- 3/8. b4
0 0 1 9/4 -23392/3 b3-9/4. b4
0 0 0 1881/1456 800/200

1 -1 -1/2 0 -595/204 b1+1.b2
0 1 1/18 0 181141/2040
0 0 1 0 -23399/200
0 0 0 1 4

1 0 -1/2 0 x1 - 70197/200
0 1 1/18 0 x2 46798/200
0 0 1 0 x3 23399/200
0 0 0 1 x4 4

x1+x2+x3-3/2x4 = 70197/200
x2+1/18-3/2x4= 46798/200
x3+9/4x4=23399/200
x4= 4
x3+9/4x4 = 70197/200
x3+9/4(4)=23399/200
x3+36/4=23399/200
x3=23399/200-36/4
x3=3

x2+1/18x3+3/8x4 = 46798/200
x2+1/18(3)+ +3/8(4) = 46798/200
x2+3/18+ +12/8= 46798/200
x2= 46798/200+51/12
x2=2
x1+x2+x3-3/2x4 = 70197/200
x1+2+3-3/2(4) = 70197/200
x1+5+12/2 = 70197/200
x1= 70197/200-7/2
x1=1


2. SISTEM PERSAMAAN LINEAR DENGAN METODE ELIMINASI GAUSS JORDAN
-2 2 1 3 17 b1x -1/2
3 -5 2 2 7
4 2 3 2 24
1 -3 2 -1 -3

1 -1 -1/2 -3/2 -17/2
3 -5 2 2 7 b2 -3.b1
4 2 3 2 24 b3 -4.b1
1 -3 2 -1 -3 b4 -b1

1 -1 -1/2 -3/2 -17/2
0 9 1/2 3/2 68/2 b2 x (1/9)
0 2 1/2 3 -170
0 -2 5/2 -10/2 - 280/51
1 -1 -1/2 -3/2 -17/2
0 1 1/18 3/8 34/9
0 2 1/2 3 -170 b3 – 4 .b2
0 -2 5/2 -10/2 - 280/51 b4 -17/2. b2

1 -1 -1/2 -3/2 -17/2
0 1 1/18 3/8 34/9
0 0 3/18 3/8 -5848/9 b3x 18/3
0 0 7/60 26/5 333217/630

1 -1 -1/2 -3/2 -17/2
0 1 1/18 3/8 34/9
0 0 1 9/4 -23392/3
0 0 7/60 26/5 630/280 b4-7/60 .b3

1 -1 -1/2 -3/2 -17/2 b1+3/2 . b4
0 1 1/18 3/8 34/9 b2- 3/8. b4
0 0 1 9/4 -23392/3 b3-9/4. b4
0 0 0 1881/1456 4

1 -1 -1/2 0 -595/204 b1+1.b2
0 1 1/18 0 181141/2040
0 0 1 0 -23399/200
0 0 0 1 4

1 -1/2 -1 0 - 70197/200 b1+1/2.b3
0 1 8 4 46798/200 b2-1/18.b3
0 0 1 0 23399/200
0 0 0 1 4

1 0 0 0 1 x1 = 1
0 -1 0 0 2 x2 = 2
0 0 1 0 3 x3 = 3
0 0 0 1 4 x4 = 4










III. DETERMINAN DENGAN DEKOMPOSISI LU
-2 2 1 3 2 0 0 0 1 -1 -1/2 -1/2
3 -5 2 2 = 3 5/3 0 0 X 0 1 63/8 9/6
2 2 3 2 4 6 63 0 0 0 1 131
5 -3 2 -1 1 4 212 30502 0 0 0 1

A11 = L11 .1 A12 = L11 . U12 A13 = L11 . U13
-2 =.L11 A12 = -2. U12 1 = -2.U13
U12 = -1 U13= 1/-2
A14 = L11.U14 A21 = L21.1 A22 = L21.U12+L22.1
3 = -2. L14 3 = L21 -5 = 3 . -1 + L22
U14= -3/2 L21 = 3 -5 = -3+L22
L22 = 5/3
A 23 = L21.U13+L22.U23 A24 = L21.U14+L22.U24
2 = 3.(- ½)+5/3.U23 2 = 3 - 3/2 + 5/3. U24
2 =-3/2-5/3.U23 2 = 9/2 + 5/3 . U24
2 = 12/15.U23 2- 9/2 = 5/3 .U24
U23 = 63/8 77/18 = 5/3.U24
U24 = 77/18 - 5/3
U24 = 9/6
U31 = L31.1 U32 = L31. U12 + L32. 1
4 = L31 2 = 4 . (-1) + L32 . 1
L31 =4 L32 = 4 + 2 .1
L32 = 6
U33 = L31 . U13 + L32 . U23.1 A 34 = L31. U14+ L32. U34
3 = 4 . – ½ + 6 . 63/8 . U23 2= 4 . 3 + 6 . 9 + 63 . U34
3 = - 4/2 + 126 + L33 2= 12 + 54 + 63
3 - 4/2 = 126 2= 129
- 1/2 = 126 U34 = 2 + 129
- 1/2 +126 =L33 U34 = 131
L33 = 63
A 41=L41.1 A 43 = L41 . U13 + L42 . 1
1 = L41 2= 1 . -2 + 4 . 63 + L43
L41 = 1 2= -2 + 252
A 42 = L41.U12 + L42 . 1 2 + 252 = L43
-3 = 1 – 1 + L42 L43 = 254
L42 = 3 + 1
L42 = 4


IV. DETERMINAN DENGAN EXPANSI KOFAKTOR / ELIMINASI PEMBENTUK MATRIK MINOR
-2X1 2X2 1X3 3X4
3X1 -5X2 2X3 2X4
4X1 2X2 3X3 2X4
1X1 -3X2 2X3 -1X4

Penyelesaian :
-2 2 1 3 + - + -
3 -5 2 2 - + - +
4 2 3 2 + - + -
1 -3 2 -1 + - + -
A. Determinan Sepanjang Garis
Det : - 2 -5 2 2 - 2 3 2 2 1 3 -5 2 - 3 3 -5 1
2 3 2 4 3 2 4 2 3 4 2 3
-3 2 1 1 2 1 1 -3 1 1 -3 2
= -2 . -5 3 2 -2 . -2 2 2 -2 . 2 2 3 -2 . 3 3 2
2 1 -3 1 -3 2 2 1
-2 . -2 4 2 -2 . 2 4 3 1 . 3 2 3 1 . 5 4 3 1. 2 4 2
1 1 1 2 -3 1 1 1 1 -3
-3 . 3 2 3 -3 . 5 4 3 -3 . 2 4 2
-3 2 1 2 1 -3
= 10 (3 - 4)+4 (2 + 8) – 4 (4+9) -6 (3-4) + 4(4-2) -4 (8-3) +3 (2+9) +5 (4-3)
+2 (-12-2) -9 (4+9) -15 (8-3) -6 (-12-2)
= 10 (-1)+4 (10)-4 (13)-6 (-1) +4 (2) -4 (5)+3 (11)+5 (1)+2 (-14)-9 (13)
-15 (5)-6 (-14)
= -10 +40-52+6+8-20+33+5-28-117-75+84
= 172
B. Determinan Sepanjang Kolom I
Det:
-2 -5 2 2 -3 2 1 3 4 2 1 3 -1 2 1 3
2 3 2 2 3 2 -5 2 2 -5 2 2
-3 2 1 -3 2 1 -3 2 1 2 3 2
= -2 . -5 3 2 -2 . -2 2 2 -2 . -3 2 2 -3 . 2 3 2
2 1 2 1 3 2 2 1
-3 . -2 1 3 -3 . -3 1 3 4 . 2 2 2 4 . -5 1 3 4 . -3 1 3
3 1 3 2 2 1 2 1 2 2
-1 . 2 2 2 -1 . -5 1 3 -1 . 2 1 3
3 2 3 2 2 2

= 10 (3 - 4)+4 (2 -4) +6 (4+6) -6 (3-4) +6 (1-4) +9 (2-9) +8 (2-4) -20 (1-6)
-12 (2-6) -2 (4-6) +5 (2-9) -2 (2-6)
= 10 (-1)+4 (-2)+6 (-2)-6 (-1) +6 (-5) +9 (-7)+8 (-2)-20 (-5)-12 (-4)-2 (-2)
+5 (-7)-2 (-4)
= -10-4-12+6-30-63-16+100+48+4-35+8
= 172
V.DETERMINAN DENGAN METODE CHIO
-2X1 2X2 1X3 3X4
3X1 -5X2 2X3 2X4
4X1 2X2 3X3 2X4
1X1 -3X2 2X3 -1X4

Penyelesaian:
-2 2 1 3
3 -5 2 2
4 2 3 2
1 -3 2 -1



det = 1/-2 4-2 -2 2 -2 1 -2 3
3 -5 3 2 3 2
-2 2 -2 1 -2 3
4 2 4 3 4 2
-2 2 -2 1 -2 3
1 -3 1 2 1 -1

Det = 1/-22 4 -7 -13
12 -10 -16
4 -5 -1

Det = 1/-2 . 1/42-2 4 -7 4 - 13
12 -10 12 - 16
4 - 7 4 - 13
4 - 5 4 - 1

Det= 1/8 44 92 Det= 1/8 ( 2112 – 736 )
8 48 Det= 1/8 ( 1376 ) = 172


3. OPERASI BARIS ELIMINASI (OBE) DENGAN REDUKSI BARIS

-4 2 4 2 1 0 0 0 b1 x -1/4
4 -3 4 2 0 1 0 0
2 3 3 2 0 0 1 0
5 6 -3 2 0 0 0 1

1 -1/2 -1 -1/2 -1/4 0 0 0
4 -3 4 2 0 1 0 0 b2 – 4 b1
2 3 3 2 0 0 1 0 b3 – 2 b1
5 6 -3 2 0 0 0 1 b4 – 5 b1


1 -1/2 -1 -1/2 -1/4 0 0 0
0 -1 8 4 1 1 0 0 b2 x (-1)
0 4 5 3 1/2 0 1 0
0 17/2 2 9/2 5/4 0 0 1

1 -1/2 -1 -1/2 -1/4 0 0 0
0 1 -8 -4 -1 -1 0 0
0 4 5 3 1/2 0 1 0 b3 – 4 b2
0 17/2 2 9/2 5/4 0 0 1 b4 - 17/2 b2

1 -1/2 -1 -1/2 -1/4 0 0 0
0 1 -8 -4 -1 -1 0 0
0 0 37 19 9/2 4 1 0 b3 x 1/37
0 0 70 77/2 39/4 17/2 0 1

1 -1/2 -1 -1/2 -1/4 0 0 0
0 1 -8 -4 -1 -1 0 0
0 0 1 19/37 9/74 4/37 1/37 0
0 0 70 77/2 39/4 17/2 0 1 b4 x 70 b3

1 -1/2 -1 -1/2 -1/4 0 0 0
0 1 -8 -4 -1 -1 0 0
0 0 1 19/37 9/74 4/37 1/37 0
0 0 0 184/74 183/148 69/74 -70/74 1 b4 x 74/189

1 -1/2 -1 -1/2 -1/4 0 0 0 b1 + 1/2 b4
0 1 -8 -4 -1 -1 0 0 b2 + 4 b4
0 0 1 19/37 9/74 4/37 1/37 0 b3 -140/189 b4
0 0 0 1 183/378 69/189 -140/189 74/189