Aljabar Linear Contoh

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

Langkah 1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Langkah 1.1

Consider the corresponding sign chart.

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

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 $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} |$

Langkah 1.4

Multiply element $a_{11}$ by its cofactor.

$1 | \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} |$

Langkah 1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} |$

Langkah 1.6

Multiply element $a_{12}$ by its cofactor.

$0 | \begin{array}{c} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} |$

Langkah 1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array} |$

Langkah 1.8

Multiply element $a_{13}$ by its cofactor.

$0 | \begin{array}{c} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array} |$

Langkah 1.9

The minor for $a_{14}$ is the determinant with row $1$ and column $4$ deleted.

$| \begin{array}{c} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array} |$

Langkah 1.10

Multiply element $a_{14}$ by its cofactor.

$0 | \begin{array}{c} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array} |$

Langkah 1.11

Add the terms together.

$1 | \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} | + 0 | \begin{array}{c} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} | + 0 | \begin{array}{c} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array} | + 0 | \begin{array}{c} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array} |$

Langkah 2

Kalikan $0$ dengan $| \begin{array}{c} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} |$ .

$1 | \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} | + 0 + 0 | \begin{array}{c} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array} | + 0 | \begin{array}{c} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array} |$

Langkah 3

Kalikan $0$ dengan $| \begin{array}{c} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array} |$ .

$1 | \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} | + 0 + 0 + 0 | \begin{array}{c} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array} |$

Langkah 4

Kalikan $0$ dengan $| \begin{array}{c} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array} |$ .

$1 | \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} | + 0 + 0 + 0$

Langkah 5

Evaluasi $| \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} |$ .

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Langkah 5.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Langkah 5.1.1

Consider the corresponding sign chart.

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

Langkah 5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Langkah 5.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 1 & 0 \\ 0 & 1 \end{array} |$

Langkah 5.1.4

Multiply element $a_{11}$ by its cofactor.

$1 | \begin{array}{c} 1 & 0 \\ 0 & 1 \end{array} |$

Langkah 5.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 0 & 0 \\ 0 & 1 \end{array} |$

Langkah 5.1.6

Multiply element $a_{12}$ by its cofactor.

$0 | \begin{array}{c} 0 & 0 \\ 0 & 1 \end{array} |$

Langkah 5.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 0 & 1 \\ 0 & 0 \end{array} |$

Langkah 5.1.8

Multiply element $a_{13}$ by its cofactor.

$0 | \begin{array}{c} 0 & 1 \\ 0 & 0 \end{array} |$

Langkah 5.1.9

Add the terms together.

$1 (1 | \begin{array}{c} 1 & 0 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 0 & 0 \\ 0 & 1 \end{array} | + 0 | \begin{array}{c} 0 & 1 \\ 0 & 0 \end{array} |) + 0 + 0 + 0$

Langkah 5.2

Kalikan $0$ dengan $| \begin{array}{c} 0 & 0 \\ 0 & 1 \end{array} |$ .

$1 (1 | \begin{array}{c} 1 & 0 \\ 0 & 1 \end{array} | + 0 + 0 | \begin{array}{c} 0 & 1 \\ 0 & 0 \end{array} |) + 0 + 0 + 0$

Langkah 5.3

Kalikan $0$ dengan $| \begin{array}{c} 0 & 1 \\ 0 & 0 \end{array} |$ .

$1 (1 | \begin{array}{c} 1 & 0 \\ 0 & 1 \end{array} | + 0 + 0) + 0 + 0 + 0$

Langkah 5.4

Evaluasi $| \begin{array}{c} 1 & 0 \\ 0 & 1 \end{array} |$ .

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Langkah 5.4.1

Determinan dari matriks $2 \times 2$ dapat dicari menggunakan rumus $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 (1 (1 \cdot 1 + 0 \cdot 0) + 0 + 0) + 0 + 0 + 0$

Langkah 5.4.2

Sederhanakan determinannya.

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Langkah 5.4.2.1

Sederhanakan setiap suku.

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Langkah 5.4.2.1.1

Kalikan $1$ dengan $1$ .

$1 (1 (1 + 0 \cdot 0) + 0 + 0) + 0 + 0 + 0$

Langkah 5.4.2.1.2

Kalikan $0$ dengan $0$ .

$1 (1 (1 + 0) + 0 + 0) + 0 + 0 + 0$

Langkah 5.4.2.2

Tambahkan $1$ dan $0$ .

$1 (1 \cdot 1 + 0 + 0) + 0 + 0 + 0$

Langkah 5.5

Sederhanakan determinannya.

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Langkah 5.5.1

Kalikan $1$ dengan $1$ .

$1 (1 + 0 + 0) + 0 + 0 + 0$

Langkah 5.5.2

Tambahkan $1$ dan $0$ .

$1 (1 + 0) + 0 + 0 + 0$

Langkah 5.5.3

Tambahkan $1$ dan $0$ .

$1 \cdot 1 + 0 + 0 + 0$

Langkah 6

Sederhanakan determinannya.

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Langkah 6.1

Kalikan $1$ dengan $1$ .

$1 + 0 + 0 + 0$

Langkah 6.2

Tambahkan $1$ dan $0$ .

$1 + 0 + 0$

Langkah 6.3

Tambahkan $1$ dan $0$ .

$1 + 0$

Langkah 6.4

Tambahkan $1$ dan $0$ .

$1$