Álgebra linear Exemplos

$[\begin{array}{c} 1 & 2 & 2 & - 4 & - 3 \\ 4 & 9 & 0 & - 2 & 3 \\ 0 & 0 & 1 & 0 & - 2 \\ 0 & 0 & 2 & 1 & 3 \\ 0 & 0 & 0 & 4 & - 4 \end{array}]$

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

$| \begin{array}{c} 9 & 0 & - 2 & 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} |$

Etapa 1.4

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

$1 | \begin{array}{c} 9 & 0 & - 2 & 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} |$

Etapa 1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

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

Etapa 1.6

Multiply element $a_{21}$ by its cofactor.

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

Etapa 1.7

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

$| \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 9 & 0 & - 2 & 3 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} |$

Etapa 1.8

Multiply element $a_{31}$ by its cofactor.

$0 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 9 & 0 & - 2 & 3 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} |$

Etapa 1.9

The minor for $a_{41}$ is the determinant with row $4$ and column $1$ deleted.

$| \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 9 & 0 & - 2 & 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 0 & 4 & - 4 \end{array} |$

Etapa 1.10

Multiply element $a_{41}$ by its cofactor.

$0 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 9 & 0 & - 2 & 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 0 & 4 & - 4 \end{array} |$

Etapa 1.11

The minor for $a_{51}$ is the determinant with row $5$ and column $1$ deleted.

$| \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 9 & 0 & - 2 & 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \end{array} |$

Etapa 1.12

Multiply element $a_{51}$ by its cofactor.

$0 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 9 & 0 & - 2 & 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \end{array} |$

Etapa 1.13

Add the terms together.

$1 | \begin{array}{c} 9 & 0 & - 2 & 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 9 & 0 & - 2 & 3 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 9 & 0 & - 2 & 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 9 & 0 & - 2 & 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \end{array} |$

Etapa 2

Multiplique $0$ por $| \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 9 & 0 & - 2 & 3 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} |$ .

$1 | \begin{array}{c} 9 & 0 & - 2 & 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 9 & 0 & - 2 & 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 9 & 0 & - 2 & 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \end{array} |$

Etapa 3

Multiplique $0$ por $| \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 9 & 0 & - 2 & 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 0 & 4 & - 4 \end{array} |$ .

$1 | \begin{array}{c} 9 & 0 & - 2 & 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 9 & 0 & - 2 & 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \end{array} |$

Etapa 4

Multiplique $0$ por $| \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 9 & 0 & - 2 & 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \end{array} |$ .

$1 | \begin{array}{c} 9 & 0 & - 2 & 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5

Avalie $| \begin{array}{c} 9 & 0 & - 2 & 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} |$ .

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Etapa 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 column $1$ by its cofactor and add.

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Etapa 5.1.1

Consider the corresponding sign chart.

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

Etapa 5.1.2

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

Etapa 5.1.3

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

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

Etapa 5.1.4

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

$9 | \begin{array}{c} 1 & 0 & - 2 \\ 2 & 1 & 3 \\ 0 & 4 & - 4 \end{array} |$

Etapa 5.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

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

Etapa 5.1.6

Multiply element $a_{21}$ by its cofactor.

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

Etapa 5.1.7

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

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

Etapa 5.1.8

Multiply element $a_{31}$ by its cofactor.

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

Etapa 5.1.9

The minor for $a_{41}$ is the determinant with row $4$ and column $1$ deleted.

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

Etapa 5.1.10

Multiply element $a_{41}$ by its cofactor.

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

Etapa 5.1.11

Add the terms together.

$1 (9 | \begin{array}{c} 1 & 0 & - 2 \\ 2 & 1 & 3 \\ 0 & 4 & - 4 \end{array} | + 0 | \begin{array}{c} 0 & - 2 & 3 \\ 2 & 1 & 3 \\ 0 & 4 & - 4 \end{array} | + 0 | \begin{array}{c} 0 & - 2 & 3 \\ 1 & 0 & - 2 \\ 0 & 4 & - 4 \end{array} | + 0 | \begin{array}{c} 0 & - 2 & 3 \\ 1 & 0 & - 2 \\ 2 & 1 & 3 \end{array} |) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.2

Multiplique $0$ por $| \begin{array}{c} 0 & - 2 & 3 \\ 2 & 1 & 3 \\ 0 & 4 & - 4 \end{array} |$ .

$1 (9 | \begin{array}{c} 1 & 0 & - 2 \\ 2 & 1 & 3 \\ 0 & 4 & - 4 \end{array} | + 0 + 0 | \begin{array}{c} 0 & - 2 & 3 \\ 1 & 0 & - 2 \\ 0 & 4 & - 4 \end{array} | + 0 | \begin{array}{c} 0 & - 2 & 3 \\ 1 & 0 & - 2 \\ 2 & 1 & 3 \end{array} |) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.3

Multiplique $0$ por $| \begin{array}{c} 0 & - 2 & 3 \\ 1 & 0 & - 2 \\ 0 & 4 & - 4 \end{array} |$ .

$1 (9 | \begin{array}{c} 1 & 0 & - 2 \\ 2 & 1 & 3 \\ 0 & 4 & - 4 \end{array} | + 0 + 0 + 0 | \begin{array}{c} 0 & - 2 & 3 \\ 1 & 0 & - 2 \\ 2 & 1 & 3 \end{array} |) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.4

Multiplique $0$ por $| \begin{array}{c} 0 & - 2 & 3 \\ 1 & 0 & - 2 \\ 2 & 1 & 3 \end{array} |$ .

$1 (9 | \begin{array}{c} 1 & 0 & - 2 \\ 2 & 1 & 3 \\ 0 & 4 & - 4 \end{array} | + 0 + 0 + 0) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.5

Avalie $| \begin{array}{c} 1 & 0 & - 2 \\ 2 & 1 & 3 \\ 0 & 4 & - 4 \end{array} |$ .

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Etapa 5.5.1

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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Etapa 5.5.1.1

Consider the corresponding sign chart.

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

Etapa 5.5.1.2

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

Etapa 5.5.1.3

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

$| \begin{array}{c} 1 & 3 \\ 4 & - 4 \end{array} |$

Etapa 5.5.1.4

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

$1 | \begin{array}{c} 1 & 3 \\ 4 & - 4 \end{array} |$

Etapa 5.5.1.5

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

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

Etapa 5.5.1.6

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

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

Etapa 5.5.1.7

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

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

Etapa 5.5.1.8

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

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

Etapa 5.5.1.9

Add the terms together.

$1 (9 (1 | \begin{array}{c} 1 & 3 \\ 4 & - 4 \end{array} | + 0 | \begin{array}{c} 2 & 3 \\ 0 & - 4 \end{array} | - 2 | \begin{array}{c} 2 & 1 \\ 0 & 4 \end{array} |) + 0 + 0 + 0) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.5.2

Multiplique $0$ por $| \begin{array}{c} 2 & 3 \\ 0 & - 4 \end{array} |$ .

$1 (9 (1 | \begin{array}{c} 1 & 3 \\ 4 & - 4 \end{array} | + 0 - 2 | \begin{array}{c} 2 & 1 \\ 0 & 4 \end{array} |) + 0 + 0 + 0) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.5.3

Avalie $| \begin{array}{c} 1 & 3 \\ 4 & - 4 \end{array} |$ .

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Etapa 5.5.3.1

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

$1 (9 (1 (1 \cdot - 4 - 4 \cdot 3) + 0 - 2 | \begin{array}{c} 2 & 1 \\ 0 & 4 \end{array} |) + 0 + 0 + 0) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.5.3.2

Simplifique o determinante.

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Etapa 5.5.3.2.1

Simplifique cada termo.

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Etapa 5.5.3.2.1.1

Multiplique $- 4$ por $1$ .

$1 (9 (1 (- 4 - 4 \cdot 3) + 0 - 2 | \begin{array}{c} 2 & 1 \\ 0 & 4 \end{array} |) + 0 + 0 + 0) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.5.3.2.1.2

Multiplique $- 4$ por $3$ .

$1 (9 (1 (- 4 - 12) + 0 - 2 | \begin{array}{c} 2 & 1 \\ 0 & 4 \end{array} |) + 0 + 0 + 0) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.5.3.2.2

Subtraia $12$ de $- 4$ .

$1 (9 (1 \cdot - 16 + 0 - 2 | \begin{array}{c} 2 & 1 \\ 0 & 4 \end{array} |) + 0 + 0 + 0) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.5.4

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

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Etapa 5.5.4.1

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

$1 (9 (1 \cdot - 16 + 0 - 2 (2 \cdot 4 + 0 \cdot 1)) + 0 + 0 + 0) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.5.4.2

Simplifique o determinante.

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Etapa 5.5.4.2.1

Simplifique cada termo.

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Etapa 5.5.4.2.1.1

Multiplique $2$ por $4$ .

$1 (9 (1 \cdot - 16 + 0 - 2 (8 + 0 \cdot 1)) + 0 + 0 + 0) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.5.4.2.1.2

Multiplique $0$ por $1$ .

$1 (9 (1 \cdot - 16 + 0 - 2 (8 + 0)) + 0 + 0 + 0) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.5.4.2.2

Some $8$ e $0$ .

$1 (9 (1 \cdot - 16 + 0 - 2 \cdot 8) + 0 + 0 + 0) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.5.5

Simplifique o determinante.

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Etapa 5.5.5.1

Simplifique cada termo.

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Etapa 5.5.5.1.1

Multiplique $- 16$ por $1$ .

$1 (9 (- 16 + 0 - 2 \cdot 8) + 0 + 0 + 0) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.5.5.1.2

Multiplique $- 2$ por $8$ .

$1 (9 (- 16 + 0 - 16) + 0 + 0 + 0) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.5.5.2

Some $- 16$ e $0$ .

$1 (9 (- 16 - 16) + 0 + 0 + 0) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.5.5.3

Subtraia $16$ de $- 16$ .

$1 (9 \cdot - 32 + 0 + 0 + 0) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.6

Simplifique o determinante.

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Etapa 5.6.1

Multiplique $9$ por $- 32$ .

$1 (- 288 + 0 + 0 + 0) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.6.2

Some $- 288$ e $0$ .

$1 (- 288 + 0 + 0) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.6.3

Some $- 288$ e $0$ .

$1 (- 288 + 0) - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 5.6.4

Some $- 288$ e $0$ .

$1 \cdot - 288 - 4 | \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} | + 0 + 0 + 0$

Etapa 6

Avalie $| \begin{array}{c} 2 & 2 & - 4 & - 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 4 & - 4 \end{array} |$ .

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Etapa 6.1

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 $1$ by its cofactor and add.

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Etapa 6.1.1

Consider the corresponding sign chart.

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

Etapa 6.1.2

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

Etapa 6.1.3

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

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

Etapa 6.1.4

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

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

Etapa 6.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

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

Etapa 6.1.6

Multiply element $a_{21}$ by its cofactor.

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

Etapa 6.1.7

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

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

Etapa 6.1.8

Multiply element $a_{31}$ by its cofactor.

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

Etapa 6.1.9

The minor for $a_{41}$ is the determinant with row $4$ and column $1$ deleted.

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

Etapa 6.1.10

Multiply element $a_{41}$ by its cofactor.

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

Etapa 6.1.11

Add the terms together.

$1 \cdot - 288 - 4 (2 | \begin{array}{c} 1 & 0 & - 2 \\ 2 & 1 & 3 \\ 0 & 4 & - 4 \end{array} | + 0 | \begin{array}{c} 2 & - 4 & - 3 \\ 2 & 1 & 3 \\ 0 & 4 & - 4 \end{array} | + 0 | \begin{array}{c} 2 & - 4 & - 3 \\ 1 & 0 & - 2 \\ 0 & 4 & - 4 \end{array} | + 0 | \begin{array}{c} 2 & - 4 & - 3 \\ 1 & 0 & - 2 \\ 2 & 1 & 3 \end{array} |) + 0 + 0 + 0$

Etapa 6.2

Multiplique $0$ por $| \begin{array}{c} 2 & - 4 & - 3 \\ 2 & 1 & 3 \\ 0 & 4 & - 4 \end{array} |$ .

$1 \cdot - 288 - 4 (2 | \begin{array}{c} 1 & 0 & - 2 \\ 2 & 1 & 3 \\ 0 & 4 & - 4 \end{array} | + 0 + 0 | \begin{array}{c} 2 & - 4 & - 3 \\ 1 & 0 & - 2 \\ 0 & 4 & - 4 \end{array} | + 0 | \begin{array}{c} 2 & - 4 & - 3 \\ 1 & 0 & - 2 \\ 2 & 1 & 3 \end{array} |) + 0 + 0 + 0$

Etapa 6.3

Multiplique $0$ por $| \begin{array}{c} 2 & - 4 & - 3 \\ 1 & 0 & - 2 \\ 0 & 4 & - 4 \end{array} |$ .

$1 \cdot - 288 - 4 (2 | \begin{array}{c} 1 & 0 & - 2 \\ 2 & 1 & 3 \\ 0 & 4 & - 4 \end{array} | + 0 + 0 + 0 | \begin{array}{c} 2 & - 4 & - 3 \\ 1 & 0 & - 2 \\ 2 & 1 & 3 \end{array} |) + 0 + 0 + 0$

Etapa 6.4

Multiplique $0$ por $| \begin{array}{c} 2 & - 4 & - 3 \\ 1 & 0 & - 2 \\ 2 & 1 & 3 \end{array} |$ .

$1 \cdot - 288 - 4 (2 | \begin{array}{c} 1 & 0 & - 2 \\ 2 & 1 & 3 \\ 0 & 4 & - 4 \end{array} | + 0 + 0 + 0) + 0 + 0 + 0$

Etapa 6.5

Avalie $| \begin{array}{c} 1 & 0 & - 2 \\ 2 & 1 & 3 \\ 0 & 4 & - 4 \end{array} |$ .

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Etapa 6.5.1

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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Etapa 6.5.1.1

Consider the corresponding sign chart.

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

Etapa 6.5.1.2

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

Etapa 6.5.1.3

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

$| \begin{array}{c} 1 & 3 \\ 4 & - 4 \end{array} |$

Etapa 6.5.1.4

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

$1 | \begin{array}{c} 1 & 3 \\ 4 & - 4 \end{array} |$

Etapa 6.5.1.5

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

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

Etapa 6.5.1.6

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

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

Etapa 6.5.1.7

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

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

Etapa 6.5.1.8

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

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

Etapa 6.5.1.9

Add the terms together.

$1 \cdot - 288 - 4 (2 (1 | \begin{array}{c} 1 & 3 \\ 4 & - 4 \end{array} | + 0 | \begin{array}{c} 2 & 3 \\ 0 & - 4 \end{array} | - 2 | \begin{array}{c} 2 & 1 \\ 0 & 4 \end{array} |) + 0 + 0 + 0) + 0 + 0 + 0$

Etapa 6.5.2

Multiplique $0$ por $| \begin{array}{c} 2 & 3 \\ 0 & - 4 \end{array} |$ .

$1 \cdot - 288 - 4 (2 (1 | \begin{array}{c} 1 & 3 \\ 4 & - 4 \end{array} | + 0 - 2 | \begin{array}{c} 2 & 1 \\ 0 & 4 \end{array} |) + 0 + 0 + 0) + 0 + 0 + 0$

Etapa 6.5.3

Avalie $| \begin{array}{c} 1 & 3 \\ 4 & - 4 \end{array} |$ .

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Etapa 6.5.3.1

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

$1 \cdot - 288 - 4 (2 (1 (1 \cdot - 4 - 4 \cdot 3) + 0 - 2 | \begin{array}{c} 2 & 1 \\ 0 & 4 \end{array} |) + 0 + 0 + 0) + 0 + 0 + 0$

Etapa 6.5.3.2

Simplifique o determinante.

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Etapa 6.5.3.2.1

Simplifique cada termo.

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Etapa 6.5.3.2.1.1

Multiplique $- 4$ por $1$ .

$1 \cdot - 288 - 4 (2 (1 (- 4 - 4 \cdot 3) + 0 - 2 | \begin{array}{c} 2 & 1 \\ 0 & 4 \end{array} |) + 0 + 0 + 0) + 0 + 0 + 0$

Etapa 6.5.3.2.1.2

Multiplique $- 4$ por $3$ .

$1 \cdot - 288 - 4 (2 (1 (- 4 - 12) + 0 - 2 | \begin{array}{c} 2 & 1 \\ 0 & 4 \end{array} |) + 0 + 0 + 0) + 0 + 0 + 0$

Etapa 6.5.3.2.2

Subtraia $12$ de $- 4$ .

$1 \cdot - 288 - 4 (2 (1 \cdot - 16 + 0 - 2 | \begin{array}{c} 2 & 1 \\ 0 & 4 \end{array} |) + 0 + 0 + 0) + 0 + 0 + 0$

Etapa 6.5.4

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

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Etapa 6.5.4.1

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

$1 \cdot - 288 - 4 (2 (1 \cdot - 16 + 0 - 2 (2 \cdot 4 + 0 \cdot 1)) + 0 + 0 + 0) + 0 + 0 + 0$

Etapa 6.5.4.2

Simplifique o determinante.

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Etapa 6.5.4.2.1

Simplifique cada termo.

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Etapa 6.5.4.2.1.1

Multiplique $2$ por $4$ .

$1 \cdot - 288 - 4 (2 (1 \cdot - 16 + 0 - 2 (8 + 0 \cdot 1)) + 0 + 0 + 0) + 0 + 0 + 0$

Etapa 6.5.4.2.1.2

Multiplique $0$ por $1$ .

$1 \cdot - 288 - 4 (2 (1 \cdot - 16 + 0 - 2 (8 + 0)) + 0 + 0 + 0) + 0 + 0 + 0$

Etapa 6.5.4.2.2

Some $8$ e $0$ .

$1 \cdot - 288 - 4 (2 (1 \cdot - 16 + 0 - 2 \cdot 8) + 0 + 0 + 0) + 0 + 0 + 0$

Etapa 6.5.5

Simplifique o determinante.

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Etapa 6.5.5.1

Simplifique cada termo.

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Etapa 6.5.5.1.1

Multiplique $- 16$ por $1$ .

$1 \cdot - 288 - 4 (2 (- 16 + 0 - 2 \cdot 8) + 0 + 0 + 0) + 0 + 0 + 0$

Etapa 6.5.5.1.2

Multiplique $- 2$ por $8$ .

$1 \cdot - 288 - 4 (2 (- 16 + 0 - 16) + 0 + 0 + 0) + 0 + 0 + 0$

Etapa 6.5.5.2

Some $- 16$ e $0$ .

$1 \cdot - 288 - 4 (2 (- 16 - 16) + 0 + 0 + 0) + 0 + 0 + 0$

Etapa 6.5.5.3

Subtraia $16$ de $- 16$ .

$1 \cdot - 288 - 4 (2 \cdot - 32 + 0 + 0 + 0) + 0 + 0 + 0$

Etapa 6.6

Simplifique o determinante.

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Etapa 6.6.1

Multiplique $2$ por $- 32$ .

$1 \cdot - 288 - 4 (- 64 + 0 + 0 + 0) + 0 + 0 + 0$

Etapa 6.6.2

Some $- 64$ e $0$ .

$1 \cdot - 288 - 4 (- 64 + 0 + 0) + 0 + 0 + 0$

Etapa 6.6.3

Some $- 64$ e $0$ .

$1 \cdot - 288 - 4 (- 64 + 0) + 0 + 0 + 0$

Etapa 6.6.4

Some $- 64$ e $0$ .

$1 \cdot - 288 - 4 \cdot - 64 + 0 + 0 + 0$

Etapa 7

Simplifique o determinante.

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Etapa 7.1

Simplifique cada termo.

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Etapa 7.1.1

Multiplique $- 288$ por $1$ .

$- 288 - 4 \cdot - 64 + 0 + 0 + 0$

Etapa 7.1.2

Multiplique $- 4$ por $- 64$ .

$- 288 + 256 + 0 + 0 + 0$

Etapa 7.2

Some $- 288$ e $256$ .

$- 32 + 0 + 0 + 0$

Etapa 7.3

Some $- 32$ e $0$ .

$- 32 + 0 + 0$

Etapa 7.4

Some $- 32$ e $0$ .

$- 32 + 0$

Etapa 7.5

Some $- 32$ e $0$ .

$- 32$