Aljabar Linear Contoh

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

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

Langkah 1

Choose the row or column with the most

0

$0$ elements. If there are no

0

$0$ elements choose any row or column. Multiply every element in row

1

$1$ by its cofactor and add.

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Langkah 1.1

Consider the corresponding sign chart.

| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Langkah 1.2

The cofactor is the minor with the sign changed if the indices match a

-

$-$ position on the sign chart.

Langkah 1.3

The minor for

a_{11}

$a_{11}$ is the determinant with row

1

$1$ and column

1

$1$ deleted.

| \begin{array}{c} - a & 1 \\ - 2 & 3 \end{array} |

$| \begin{array}{c} - a & 1 \\ - 2 & 3 \end{array} |$

Langkah 1.4

Multiply element

a_{11}

$a_{11}$ by its cofactor.

0 | \begin{array}{c} - a & 1 \\ - 2 & 3 \end{array} |

$0 | \begin{array}{c} - a & 1 \\ - 2 & 3 \end{array} |$

Langkah 1.5

The minor for

a_{12}

$a_{12}$ is the determinant with row

1

$1$ and column

2

$2$ deleted.

| \begin{array}{c} 3 & 1 \\ 1 & 3 \end{array} |

$| \begin{array}{c} 3 & 1 \\ 1 & 3 \end{array} |$

Langkah 1.6

Multiply element

a_{12}

$a_{12}$ by its cofactor.

1 | \begin{array}{c} 3 & 1 \\ 1 & 3 \end{array} |

$1 | \begin{array}{c} 3 & 1 \\ 1 & 3 \end{array} |$

Langkah 1.7

The minor for

a_{13}

$a_{13}$ is the determinant with row

1

$1$ and column

3

$3$ deleted.

| \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |

$| \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |$

Langkah 1.8

Multiply element

a_{13}

$a_{13}$ by its cofactor.

a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |

$a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |$

Langkah 1.9

Add the terms together.

0 | \begin{array}{c} - a & 1 \\ - 2 & 3 \end{array} | + 1 | \begin{array}{c} 3 & 1 \\ 1 & 3 \end{array} | + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |

$0 | \begin{array}{c} - a & 1 \\ - 2 & 3 \end{array} | + 1 | \begin{array}{c} 3 & 1 \\ 1 & 3 \end{array} | + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |$

0 | \begin{array}{c} - a & 1 \\ - 2 & 3 \end{array} | + 1 | \begin{array}{c} 3 & 1 \\ 1 & 3 \end{array} | + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |

$0 | \begin{array}{c} - a & 1 \\ - 2 & 3 \end{array} | + 1 | \begin{array}{c} 3 & 1 \\ 1 & 3 \end{array} | + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |$

Langkah 2

Kalikan

0

$0$ dengan

| \begin{array}{c} - a & 1 \\ - 2 & 3 \end{array} |

$| \begin{array}{c} - a & 1 \\ - 2 & 3 \end{array} |$ .

0 + 1 | \begin{array}{c} 3 & 1 \\ 1 & 3 \end{array} | + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |

$0 + 1 | \begin{array}{c} 3 & 1 \\ 1 & 3 \end{array} | + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |$

Langkah 3

Evaluasi

| \begin{array}{c} 3 & 1 \\ 1 & 3 \end{array} |

$| \begin{array}{c} 3 & 1 \\ 1 & 3 \end{array} |$ .

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Langkah 3.1

Determinan dari matriks

2 \times 2

$2 \times 2$ dapat dicari menggunakan rumus

| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

0 + 1 (3 \cdot 3 - 1 \cdot 1) + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |

$0 + 1 (3 \cdot 3 - 1 \cdot 1) + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |$

Langkah 3.2

Sederhanakan determinannya.

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Langkah 3.2.1

Sederhanakan setiap suku.

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Langkah 3.2.1.1

Kalikan

3

$3$ dengan

3

$3$ .

0 + 1 (9 - 1 \cdot 1) + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |

$0 + 1 (9 - 1 \cdot 1) + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |$

Langkah 3.2.1.2

Kalikan

- 1

$- 1$ dengan

1

$1$ .

0 + 1 (9 - 1) + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |

$0 + 1 (9 - 1) + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |$

0 + 1 (9 - 1) + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |

$0 + 1 (9 - 1) + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |$

Langkah 3.2.2

Kurangi

1

$1$ dengan

9

$9$ .

0 + 1 \cdot 8 + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |

$0 + 1 \cdot 8 + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |$

0 + 1 \cdot 8 + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |

$0 + 1 \cdot 8 + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |$

0 + 1 \cdot 8 + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |

$0 + 1 \cdot 8 + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |$

Langkah 4

Evaluasi

| \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |

$| \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |$ .

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Langkah 4.1

Determinan dari matriks

2 \times 2

$2 \times 2$ dapat dicari menggunakan rumus

| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

0 + 1 \cdot 8 + a (3 \cdot - 2 - - a)

$0 + 1 \cdot 8 + a (3 \cdot - 2 - - a)$

Langkah 4.2

Sederhanakan setiap suku.

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Langkah 4.2.1

Kalikan

3

$3$ dengan

- 2

$- 2$ .

0 + 1 \cdot 8 + a (- 6 - - a)

$0 + 1 \cdot 8 + a (- 6 - - a)$

Langkah 4.2.2

Kalikan

- - a

$- - a$ .

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Langkah 4.2.2.1

Kalikan

- 1

$- 1$ dengan

- 1

$- 1$ .

0 + 1 \cdot 8 + a (- 6 + 1 a)

$0 + 1 \cdot 8 + a (- 6 + 1 a)$

Langkah 4.2.2.2

Kalikan

a

$a$ dengan

1

$1$ .

0 + 1 \cdot 8 + a (- 6 + a)

$0 + 1 \cdot 8 + a (- 6 + a)$

0 + 1 \cdot 8 + a (- 6 + a)

$0 + 1 \cdot 8 + a (- 6 + a)$

0 + 1 \cdot 8 + a (- 6 + a)

$0 + 1 \cdot 8 + a (- 6 + a)$

0 + 1 \cdot 8 + a (- 6 + a)

$0 + 1 \cdot 8 + a (- 6 + a)$

Langkah 5

Sederhanakan determinannya.

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Langkah 5.1

Tambahkan

0

$0$ dan

1 \cdot 8

$1 \cdot 8$ .

1 \cdot 8 + a (- 6 + a)

$1 \cdot 8 + a (- 6 + a)$

Langkah 5.2

Sederhanakan setiap suku.

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Langkah 5.2.1

Kalikan

8

$8$ dengan

1

$1$ .

8 + a (- 6 + a)

$8 + a (- 6 + a)$

Langkah 5.2.2

Terapkan sifat distributif.

8 + a \cdot - 6 + a \cdot a

$8 + a \cdot - 6 + a \cdot a$

Langkah 5.2.3

Pindahkan

- 6

$- 6$ ke sebelah kiri

a

$a$ .

8 - 6 \cdot a + a \cdot a

$8 - 6 \cdot a + a \cdot a$

Langkah 5.2.4

Kalikan

a

$a$ dengan

a

$a$ .

8 - 6 a + a^{2}

$8 - 6 a + a^{2}$

8 - 6 a + a^{2}

$8 - 6 a + a^{2}$

8 - 6 a + a^{2}

$8 - 6 a + a^{2}$