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Eleminasi Gauss

Soal

1. Sistem Persamaan Linear

{x1+x2+x3+x4+x5=152x1+3x2+2x3+2x4+2x5=31x2+2x3+x4+x5=10x1+x2+x3+2x4+2x5=212x5=4\begin{cases} x_1 + x_2 + x_3 + x_4 + x_5 = 15 \\ 2x_1 + 3x_2 + 2x_3 + 2x_4 + 2x_5 = 31 \\ x_2 + 2x_3 + x_4 + x_5 = 10 \\ x_1 + x_2 + x_3 + 2x_4 + 2x_5 = 21 \\ 2x_5 = 4 \end{cases}
(1)

2. Matriks Augmentasi Awal

1111115232223101211101112221000024\begin{array}{ccccc|c} 1 & 1 & 1 & 1 & 1 & 15 \\ 2 & 3 & 2 & 2 & 2 & 31 \\ 0 & 1 & 2 & 1 & 1 & 10 \\ 1 & 1 & 1 & 2 & 2 & 21 \\ 0 & 0 & 0 & 0 & 2 & 4 \end{array}
(2)

Penyelesaian

1. Membersihkan Kolom 1 (Bawah Pivot x1x_1)

Operasi :

R2=R2−2R1R4=R4−R1\begin{matrix} R_2 = R_2-2R_1\\ R_4 = R_4-R_1 \end{matrix}
(3)

Hasil :

11111150100010121110000116000024\begin{array}{ccccc|c} 1 & 1 & 1 & 1 & 1 & 15 \\ 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 2 & 1 & 1 & 10 \\ 0 & 0 & 0 & 1 & 1 & 6 \\ 0 & 0 & 0 & 0 & 2 & 4 \end{array}
(4)

2. Membersihkan Kolom 2 (Bawah Pivot x2x_2)

Operasi :

R3=R3−R2\begin{matrix} R_3 = R_3-R_2 \end{matrix}
(5)

Hasil :

1111115010001002119000116000024\begin{array}{ccccc|c} 1 & 1 & 1 & 1 & 1 & 15 \\ 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 2 & 1 & 1 & 9 \\ 0 & 0 & 0 & 1 & 1 & 6 \\ 0 & 0 & 0 & 0 & 2 & 4 \end{array}
(6)

3. Membuat pivot terakhir = 1

Operasi :

R5=12R5\begin{matrix} R_5 = \frac{1}{2}R_5 \end{matrix}
(7)

Hasil :

1111115010001002119000116000012\begin{array}{ccccc|c} 1 & 1 & 1 & 1 & 1 & 15 \\ 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 2 & 1 & 1 & 9 \\ 0 & 0 & 0 & 1 & 1 & 6 \\ 0 & 0 & 0 & 0 & 1 & 2 \end{array}
(8)

4. Menghilangkan angka di atas pivot x5x_5

Operasi :

R4=R4−R5R3=R3−R5R1=R1−R5\begin{matrix} R_4 = R_4 - R_5\\ R_3 = R_3 - R_5\\ R_1 = R_1 - R_5 \end{matrix}
(9)

Hasil :

1111013010001002107000104000012\begin{array}{ccccc|c} 1 & 1 & 1 & 1 & 0 & 13 \\ 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 2 & 1 & 0 & 7 \\ 0 & 0 & 0 & 1 & 0 & 4 \\ 0 & 0 & 0 & 0 & 1 & 2 \end{array}
(10)

5. Menghilangkan angka di atas pivot x4x_4

Operasi :

R1=R1−R4R3=R3−R4\begin{matrix} R_1 = R_1 - R_4\\ R_3 = R_3 - R_4 \end{matrix}
(11)

Hasil :

111009010001002003000104000012\begin{array}{ccccc|c} 1 & 1 & 1 & 0 & 0 & 9 \\ 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 2 & 0 & 0 & 3 \\ 0 & 0 & 0 & 1 & 0 & 4 \\ 0 & 0 & 0 & 0 & 1 & 2 \end{array}
(12)

6. Membuat pivot x3x_3 = 1

Operasi :

R3=12R3\begin{matrix} R_3 = \frac{1}{2}R_3 \end{matrix}
(13)

Hasil :

1110090100010010032000104000012\begin{array}{ccccc|c} 1 & 1 & 1 & 0 & 0 & 9 \\ 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & \frac{3}{2} \\ 0 & 0 & 0 & 1 & 0 & 4 \\ 0 & 0 & 0 & 0 & 1 & 2 \end{array}
(14)

7. Menghilangkan angka di atas pivot x3x_3

Operasi :

R1=R1−R3\begin{matrix} R_1 = R_1 - R_3\\ \end{matrix}
(15)

Hasil :

110001520100010010032000104000012\begin{array}{ccccc|c} 1 & 1 & 0 & 0 & 0 & \frac {15}{2} \\ 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & \frac{3}{2} \\ 0 & 0 & 0 & 1 & 0 & 4 \\ 0 & 0 & 0 & 0 & 1 & 2 \end{array}
(16)

8. Menghilangkan angka di atas pivot x2x_2

Operasi :

R1=R1−R2\begin{matrix} R_1 = R_1 - R_2 \end{matrix}
(17)

Hasil :

100001320100010010032000104000012\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & 0 & \frac {13}{2} \\ 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & \frac {3}{2} \\ 0 & 0 & 0 & 1 & 0 & 4 \\ 0 & 0 & 0 & 0 & 1 & 2 \end{array}
(18)

9. Hasil Akhir

(x1=132,x2=1,x3=32,x4=4,x5=2)(x_1 = \frac {13}{2}, x_2 = 1, x_3 = \frac {3}{2},x_4 = 4, x_5 = 2 )
(19)

10. Uji Coba dengan Sage cell

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Eleminasi Gauss
Persamaan Linier (Semester 2)