Álgebra linear Exemplos

$[\begin{array}{c} 2 & 4 & 0 \\ 4 & 6 & 3 \\ - 6 & - 10 & 0 \end{array}]$

Etapa 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 column $3$ by its cofactor and add.

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Etapa 1.1

Consider the corresponding sign chart.

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

Etapa 1.2

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

Etapa 1.3

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

$| \begin{array}{c} 4 & 6 \\ - 6 & - 10 \end{array} |$

Etapa 1.4

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

$0 | \begin{array}{c} 4 & 6 \\ - 6 & - 10 \end{array} |$

Etapa 1.5

The minor for $a_{23}$ is the determinant with row $2$ and column $3$ deleted.

$| \begin{array}{c} 2 & 4 \\ - 6 & - 10 \end{array} |$

Etapa 1.6

Multiply element $a_{23}$ by its cofactor.

$- 3 | \begin{array}{c} 2 & 4 \\ - 6 & - 10 \end{array} |$

Etapa 1.7

The minor for $a_{33}$ is the determinant with row $3$ and column $3$ deleted.

$| \begin{array}{c} 2 & 4 \\ 4 & 6 \end{array} |$

Etapa 1.8

Multiply element $a_{33}$ by its cofactor.

$0 | \begin{array}{c} 2 & 4 \\ 4 & 6 \end{array} |$

Etapa 1.9

Add the terms together.

$0 | \begin{array}{c} 4 & 6 \\ - 6 & - 10 \end{array} | - 3 | \begin{array}{c} 2 & 4 \\ - 6 & - 10 \end{array} | + 0 | \begin{array}{c} 2 & 4 \\ 4 & 6 \end{array} |$

Etapa 2

Multiplique $0$ por $| \begin{array}{c} 4 & 6 \\ - 6 & - 10 \end{array} |$ .

$0 - 3 | \begin{array}{c} 2 & 4 \\ - 6 & - 10 \end{array} | + 0 | \begin{array}{c} 2 & 4 \\ 4 & 6 \end{array} |$

Etapa 3

Multiplique $0$ por $| \begin{array}{c} 2 & 4 \\ 4 & 6 \end{array} |$ .

$0 - 3 | \begin{array}{c} 2 & 4 \\ - 6 & - 10 \end{array} | + 0$

Etapa 4

Avalie $| \begin{array}{c} 2 & 4 \\ - 6 & - 10 \end{array} |$ .

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Etapa 4.1

O determinante de uma matriz $2 \times 2$ pode ser encontrado ao usar a fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$0 - 3 (2 \cdot - 10 - (- 6 \cdot 4)) + 0$

Etapa 4.2

Simplifique o determinante.

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Etapa 4.2.1

Simplifique cada termo.

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Etapa 4.2.1.1

Multiplique $2$ por $- 10$ .

$0 - 3 (- 20 - (- 6 \cdot 4)) + 0$

Etapa 4.2.1.2

Multiplique $- (- 6 \cdot 4)$ .

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Etapa 4.2.1.2.1

Multiplique $- 6$ por $4$ .

$0 - 3 (- 20 - - 24) + 0$

Etapa 4.2.1.2.2

Multiplique $- 1$ por $- 24$ .

$0 - 3 (- 20 + 24) + 0$

Etapa 4.2.2

Some $- 20$ e $24$ .

$0 - 3 \cdot 4 + 0$

Etapa 5

Simplifique o determinante.

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Etapa 5.1

Multiplique $- 3$ por $4$ .

$0 - 12 + 0$

Etapa 5.2

Subtraia $12$ de $0$ .

$- 12 + 0$

Etapa 5.3

Some $- 12$ e $0$ .

$- 12$