Álgebra Exemplos

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

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

$| \begin{array}{c} - 1 & - 2 \\ 5 & 1 \end{array} |$

Etapa 1.4

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

$- 1 | \begin{array}{c} - 1 & - 2 \\ 5 & 1 \end{array} |$

Etapa 1.5

The minor for $a_{22}$ is the determinant with row $2$ and column $2$ deleted.

$| \begin{array}{c} - 4 & 2 \\ 5 & 1 \end{array} |$

Etapa 1.6

Multiply element $a_{22}$ by its cofactor.

$0 | \begin{array}{c} - 4 & 2 \\ 5 & 1 \end{array} |$

Etapa 1.7

The minor for $a_{32}$ is the determinant with row $3$ and column $2$ deleted.

$| \begin{array}{c} - 4 & 2 \\ - 1 & - 2 \end{array} |$

Etapa 1.8

Multiply element $a_{32}$ by its cofactor.

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

Etapa 1.9

Add the terms together.

$- 1 | \begin{array}{c} - 1 & - 2 \\ 5 & 1 \end{array} | + 0 | \begin{array}{c} - 4 & 2 \\ 5 & 1 \end{array} | + 0 | \begin{array}{c} - 4 & 2 \\ - 1 & - 2 \end{array} |$

Etapa 2

Multiplique $0$ por $| \begin{array}{c} - 4 & 2 \\ 5 & 1 \end{array} |$ .

$- 1 | \begin{array}{c} - 1 & - 2 \\ 5 & 1 \end{array} | + 0 + 0 | \begin{array}{c} - 4 & 2 \\ - 1 & - 2 \end{array} |$

Etapa 3

Multiplique $0$ por $| \begin{array}{c} - 4 & 2 \\ - 1 & - 2 \end{array} |$ .

$- 1 | \begin{array}{c} - 1 & - 2 \\ 5 & 1 \end{array} | + 0 + 0$

Etapa 4

Avalie $| \begin{array}{c} - 1 & - 2 \\ 5 & 1 \end{array} |$ .

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Etapa 4.1

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

$- 1 (- 1 \cdot 1 - 5 \cdot - 2) + 0 + 0$

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 $- 1$ por $1$ .

$- 1 (- 1 - 5 \cdot - 2) + 0 + 0$

Etapa 4.2.1.2

Multiplique $- 5$ por $- 2$ .

$- 1 (- 1 + 10) + 0 + 0$

Etapa 4.2.2

Some $- 1$ e $10$ .

$- 1 \cdot 9 + 0 + 0$

Etapa 5

Simplifique o determinante.

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Etapa 5.1

Multiplique $- 1$ por $9$ .

$- 9 + 0 + 0$

Etapa 5.2

Some $- 9$ e $0$ .

$- 9 + 0$

Etapa 5.3

Some $- 9$ e $0$ .

$- 9$