Aljabar Linear Contoh

x + y + z = 2

$x + y + z = 2$ ,

4 x + 5 y + z = 12

$4 x + 5 y + z = 12$ ,

2 x = - 4

$2 x = - 4$

Langkah 1

Tulis sistem persamaan tersebut dalam bentuk matriks.

[\begin{array}{c} 1 & 1 & 1 & 2 \\ 4 & 5 & 1 & 12 \\ 2 & 0 & 0 & - 4 \end{array}]

$[\begin{array}{c} 1 & 1 & 1 & 2 \\ 4 & 5 & 1 & 12 \\ 2 & 0 & 0 & - 4 \end{array}]$

Langkah 2

Tentukan bentuk eselon baris yang dikurangi.

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Langkah 2.1

Perform the row operation

R_{2} = R_{2} - 4 R_{1}

$R_{2} = R_{2} - 4 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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Langkah 2.1.1

Perform the row operation

R_{2} = R_{2} - 4 R_{1}

$R_{2} = R_{2} - 4 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

[\begin{array}{c} 1 & 1 & 1 & 2 \\ 4 - 4 \cdot 1 & 5 - 4 \cdot 1 & 1 - 4 \cdot 1 & 12 - 4 \cdot 2 \\ 2 & 0 & 0 & - 4 \end{array}]

$[\begin{array}{c} 1 & 1 & 1 & 2 \\ 4 - 4 \cdot 1 & 5 - 4 \cdot 1 & 1 - 4 \cdot 1 & 12 - 4 \cdot 2 \\ 2 & 0 & 0 & - 4 \end{array}]$

Langkah 2.1.2

Sederhanakan

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ 2 & 0 & 0 & - 4 \end{array}]

$[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ 2 & 0 & 0 & - 4 \end{array}]$

[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ 2 & 0 & 0 & - 4 \end{array}]

$[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ 2 & 0 & 0 & - 4 \end{array}]$

Langkah 2.2

Perform the row operation

R_{3} = R_{3} - 2 R_{1}

$R_{3} = R_{3} - 2 R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

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Langkah 2.2.1

Perform the row operation

R_{3} = R_{3} - 2 R_{1}

$R_{3} = R_{3} - 2 R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ 2 - 2 \cdot 1 & 0 - 2 \cdot 1 & 0 - 2 \cdot 1 & - 4 - 2 \cdot 2 \end{array}]

$[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ 2 - 2 \cdot 1 & 0 - 2 \cdot 1 & 0 - 2 \cdot 1 & - 4 - 2 \cdot 2 \end{array}]$

Langkah 2.2.2

Sederhanakan

R_{3}

$R_{3}$ .

[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ 0 & - 2 & - 2 & - 8 \end{array}]

$[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ 0 & - 2 & - 2 & - 8 \end{array}]$

[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ 0 & - 2 & - 2 & - 8 \end{array}]

$[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ 0 & - 2 & - 2 & - 8 \end{array}]$

Langkah 2.3

Perform the row operation

R_{3} = R_{3} + 2 R_{2}

$R_{3} = R_{3} + 2 R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

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Langkah 2.3.1

Perform the row operation

R_{3} = R_{3} + 2 R_{2}

$R_{3} = R_{3} + 2 R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ 0 + 2 \cdot 0 & - 2 + 2 \cdot 1 & - 2 + 2 \cdot - 3 & - 8 + 2 \cdot 4 \end{array}]

$[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ 0 + 2 \cdot 0 & - 2 + 2 \cdot 1 & - 2 + 2 \cdot - 3 & - 8 + 2 \cdot 4 \end{array}]$

Langkah 2.3.2

Sederhanakan

R_{3}

$R_{3}$ .

[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ 0 & 0 & - 8 & 0 \end{array}]

$[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ 0 & 0 & - 8 & 0 \end{array}]$

[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ 0 & 0 & - 8 & 0 \end{array}]

$[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ 0 & 0 & - 8 & 0 \end{array}]$

Langkah 2.4

Multiply each element of

R_{3}

$R_{3}$ by

- \frac{1}{8}

$- \frac{1}{8}$ to make the entry at

3, 3

$3, 3$ a

1

$1$ .

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Langkah 2.4.1

Multiply each element of

R_{3}

$R_{3}$ by

- \frac{1}{8}

$- \frac{1}{8}$ to make the entry at

3, 3

$3, 3$ a

1

$1$ .

[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ - \frac{1}{8} \cdot 0 & - \frac{1}{8} \cdot 0 & - \frac{1}{8} \cdot - 8 & - \frac{1}{8} \cdot 0 \end{array}]

$[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ - \frac{1}{8} \cdot 0 & - \frac{1}{8} \cdot 0 & - \frac{1}{8} \cdot - 8 & - \frac{1}{8} \cdot 0 \end{array}]$

Langkah 2.4.2

Sederhanakan

R_{3}

$R_{3}$ .

[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ 0 & 0 & 1 & 0 \end{array}]$

[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & - 3 & 4 \\ 0 & 0 & 1 & 0 \end{array}]$

Langkah 2.5

Perform the row operation

R_{2} = R_{2} + 3 R_{3}

$R_{2} = R_{2} + 3 R_{3}$ to make the entry at

2, 3

$2, 3$ a

0

$0$ .

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Langkah 2.5.1

Perform the row operation

R_{2} = R_{2} + 3 R_{3}

$R_{2} = R_{2} + 3 R_{3}$ to make the entry at

2, 3

$2, 3$ a

0

$0$ .

[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 + 3 \cdot 0 & 1 + 3 \cdot 0 & - 3 + 3 \cdot 1 & 4 + 3 \cdot 0 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 + 3 \cdot 0 & 1 + 3 \cdot 0 & - 3 + 3 \cdot 1 & 4 + 3 \cdot 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Langkah 2.5.2

Sederhanakan

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 \end{array}]$

[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 & 1 & 1 & 2 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 \end{array}]$

Langkah 2.6

Perform the row operation

R_{1} = R_{1} - R_{3}

$R_{1} = R_{1} - R_{3}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

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Langkah 2.6.1

Perform the row operation

R_{1} = R_{1} - R_{3}

$R_{1} = R_{1} - R_{3}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

[\begin{array}{c} 1 - 0 & 1 - 0 & 1 - 1 & 2 - 0 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 - 0 & 1 - 0 & 1 - 1 & 2 - 0 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 \end{array}]$

Langkah 2.6.2

Sederhanakan

R_{1}

$R_{1}$ .

[\begin{array}{c} 1 & 1 & 0 & 2 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 & 1 & 0 & 2 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 \end{array}]$

[\begin{array}{c} 1 & 1 & 0 & 2 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 & 1 & 0 & 2 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 \end{array}]$

Langkah 2.7

Perform the row operation

R_{1} = R_{1} - R_{2}

$R_{1} = R_{1} - R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

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Langkah 2.7.1

Perform the row operation

R_{1} = R_{1} - R_{2}

$R_{1} = R_{1} - R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

[\begin{array}{c} 1 - 0 & 1 - 1 & 0 - 0 & 2 - 4 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 - 0 & 1 - 1 & 0 - 0 & 2 - 4 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 \end{array}]$

Langkah 2.7.2

Sederhanakan

R_{1}

$R_{1}$ .

[\begin{array}{c} 1 & 0 & 0 & - 2 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 & 0 & 0 & - 2 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 \end{array}]$

[\begin{array}{c} 1 & 0 & 0 & - 2 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 & 0 & 0 & - 2 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 \end{array}]$

[\begin{array}{c} 1 & 0 & 0 & - 2 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 & 0 & 0 & - 2 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 \end{array}]$

Langkah 3

Gunakan matriks hasil untuk menyatakan penyelesaian akhir untuk sistem persamaan tersebut.

x = - 2

$x = - 2$

y = 4

$y = 4$

z = 0

$z = 0$

Langkah 4

Penyelesaiannya adalah himpunan pasangan terurut yang membuat sistem tersebut benar.

(- 2, 4, 0)

$(- 2, 4, 0)$

Langkah 5

Menguraikan vektor penyelesaian dengan menata ulang setiap persamaan yang diwakilkan dalam bentuk pengurangan-baris dari matriks imbuhan dengan menyelesaikan variabel terikat di setiap baris sehingga menghasilkan persamaan vektor.

X = [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} - 2 \\ 4 \\ 0 \end{array}]

$X = [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} - 2 \\ 4 \\ 0 \end{array}]$