Sistem Persamaan Linear

Januari 13, 2014

Contoh Soal

3x3 + 5x1 = 7
2x2 + 3x3 = 9
4x2 = 11

a. Ubahlah sistem persamaan linear tersebut menjadi matriks A . x = B
b. Carilah x1, x2 dan x3 menggunakan metode Invers, metode Crammer, metode Gauss, dan metode Gauss-Jordan !

Jawaban :
a.

A x B
b.

Metode Invers

A-1 = adj A x B

det A

                 = -12     12     -6           7
   0     0      -15     x    9
   0     -20      10           11
                           -60
                 = -84 + 108     + (-66)
                       0   +   0       + (-165)
                       0   + (-180) + 110
                                -60
                 = -42                -42 / -60
                     -165      atau      -165 / -60
                     -70                -70 / -60

                     -60

          x1       7 / 10
x2   = 11 / 4
x3       7 / 6

Metode Crammer

det A =

5 0

0 2

0 4

                   = (5 x 2 x 0) + (0 x 3 x 0) + (3 x 0 x 4) - (3 x 2 x 0) - (5 x 3 x 4) - (0 x 0 x 0)

= 0 + 0 + 0 - 0 - 60 - 0

= -60

det x1 =

7 0

9 2

11 4

= (7 x 2 x 0) + (0 x 3 11) + (3 x 9 x 4) - (3 x 2 x 11) - (7 x 3 x 4) - (0 x 9 x 0)

= 0 + 0 + 108 - 66 - 84 - 0

= -42

det x2 =

5 7

0 9

0 11

= (5 x 9 x 0) + (7 x 3 x 0) + (3 x 0 x 11) - (3 x 9 x 0) - (5 x 3 x 11) - (7 x 0 x 0)

= 0 + 0 + 0 - 0 - 165 - 0

= -165

det x3 =

5 0

0   2

0   4



= (5 x 2 x 11) + (0 x 9 x 0) + (7 x 0 x 4) - (7 x 2 x 0) - (5 x 9 x 4) - (0 x 0 x 11)
                    = 110 + 0 + 0 - 0 - 180 - 0
                    = -70
Solusi x1 = det x1
                  det A
               = -42 / -60
               = 7 / 10

          x2 = det x2
                  det A
               = -165 / -60
               = 11 / 4

          x3 = det x3
                  det A
               = -70 / -60
               = 7 / 6

Metode Gauss

H1 (1/5)

H2 (1/2) H32 (-4)

H3 (-1/2)

Persamaan 3 = x3 = 7 / 6

Persamaan 2 = x2 + 3 / 2 (x3) = 9 / 2
x2 + 3 / 2 (7 / 6) = 9 / 2
      x2 + 21 / 12 = 9 / 2
      x2 = 9 / 2 - 21 / 12
    x2 = 54 - 21
                                   12
                        x2 = 33 / 12
                        x2 = 11 / 4

Persamaan 1 = x1 + 0 (x2) + 3 / 5 (x3) = 7 / 5
                        x1 + 0 (11 / 4) + 3 / 5 (7 / 6) = 7 / 5
                        x1 + 0 + 21 / 30 = 7 / 5
                        x1 = 7 / 5 - 21 / 30
                        x1 = 42 - 21
                                    30
                        x1 = 21 / 30
                        x1 = 7 / 10