Álgebra linear Exemplos

$[\begin{array}{c} \frac{1}{11} & \frac{1}{11} & - \frac{5}{11} \\ \frac{1}{11} & \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & - \frac{10}{11} & \frac{1}{22} \end{array}]$

Etapa 1

Find the determinant.

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Etapa 1.1

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 1.1.1

Consider the corresponding sign chart.

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

Etapa 1.1.2

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

Etapa 1.1.3

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

$| \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ - \frac{10}{11} & \frac{1}{22} \end{array} |$

Etapa 1.1.4

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

$\frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ - \frac{10}{11} & \frac{1}{22} \end{array} |$

Etapa 1.1.5

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

$| \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} |$

Etapa 1.1.6

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

$- \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} |$

Etapa 1.1.7

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

$| \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.1.8

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

$- \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.1.9

Add the terms together.

$\frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ - \frac{10}{11} & \frac{1}{22} \end{array} | - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2

Avalie $| \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ - \frac{10}{11} & \frac{1}{22} \end{array} |$ .

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Etapa 1.2.1

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

$\frac{1}{11} (\frac{1}{11} \cdot \frac{1}{22} - (- \frac{10}{11} \cdot \frac{1}{22})) - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2.2

Simplifique o determinante.

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Etapa 1.2.2.1

Simplifique cada termo.

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Etapa 1.2.2.1.1

Multiplique $\frac{1}{11} \cdot \frac{1}{22}$ .

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Etapa 1.2.2.1.1.1

Multiplique $\frac{1}{11}$ por $\frac{1}{22}$ .

$\frac{1}{11} (\frac{1}{11 \cdot 22} - (- \frac{10}{11} \cdot \frac{1}{22})) - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2.2.1.1.2

Multiplique $11$ por $22$ .

$\frac{1}{11} (\frac{1}{242} - (- \frac{10}{11} \cdot \frac{1}{22})) - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2.2.1.2

Cancele o fator comum de $2$ .

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Etapa 1.2.2.1.2.1

Mova o negativo de maior ordem em $- \frac{10}{11}$ para o numerador.

$\frac{1}{11} (\frac{1}{242} - (\frac{- 10}{11} \cdot \frac{1}{22})) - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2.2.1.2.2

Fatore $2$ de $- 10$ .

$\frac{1}{11} (\frac{1}{242} - (\frac{2 (- 5)}{11} \cdot \frac{1}{22})) - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2.2.1.2.3

Fatore $2$ de $22$ .

$\frac{1}{11} (\frac{1}{242} - (\frac{2 \cdot - 5}{11} \cdot \frac{1}{2 \cdot 11})) - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2.2.1.2.4

Cancele o fator comum.

Etapa 1.2.2.1.2.5

Reescreva a expressão.

$\frac{1}{11} (\frac{1}{242} - (\frac{- 5}{11} \cdot \frac{1}{11})) - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2.2.1.3

Multiplique $\frac{- 5}{11}$ por $\frac{1}{11}$ .

$\frac{1}{11} (\frac{1}{242} - \frac{- 5}{11 \cdot 11}) - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2.2.1.4

Multiplique $11$ por $11$ .

$\frac{1}{11} (\frac{1}{242} - \frac{- 5}{121}) - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2.2.1.5

Mova o número negativo para a frente da fração.

$\frac{1}{11} (\frac{1}{242} - - \frac{5}{121}) - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2.2.1.6

Multiplique $- - \frac{5}{121}$ .

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Etapa 1.2.2.1.6.1

Multiplique $- 1$ por $- 1$ .

$\frac{1}{11} (\frac{1}{242} + 1 (\frac{5}{121})) - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2.2.1.6.2

Multiplique $\frac{5}{121}$ por $1$ .

$\frac{1}{11} (\frac{1}{242} + \frac{5}{121}) - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2.2.2

Para escrever $\frac{5}{121}$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$\frac{1}{11} (\frac{1}{242} + \frac{5}{121} \cdot \frac{2}{2}) - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2.2.3

Escreva cada expressão com um denominador comum de $242$ , multiplicando cada um por um fator apropriado de $1$ .

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Etapa 1.2.2.3.1

Multiplique $\frac{5}{121}$ por $\frac{2}{2}$ .

$\frac{1}{11} (\frac{1}{242} + \frac{5 \cdot 2}{121 \cdot 2}) - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2.2.3.2

Multiplique $121$ por $2$ .

$\frac{1}{11} (\frac{1}{242} + \frac{5 \cdot 2}{242}) - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2.2.4

Combine os numeradores em relação ao denominador comum.

$\frac{1}{11} \cdot \frac{1 + 5 \cdot 2}{242} - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2.2.5

Simplifique o numerador.

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Etapa 1.2.2.5.1

Multiplique $5$ por $2$ .

$\frac{1}{11} \cdot \frac{1 + 10}{242} - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2.2.5.2

Some $1$ e $10$ .

$\frac{1}{11} \cdot \frac{11}{242} - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2.2.6

Cancele o fator comum de $11$ e $242$ .

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Etapa 1.2.2.6.1

Fatore $11$ de $11$ .

$\frac{1}{11} \cdot \frac{11 (1)}{242} - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2.2.6.2

Cancele os fatores comuns.

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Etapa 1.2.2.6.2.1

Fatore $11$ de $242$ .

$\frac{1}{11} \cdot \frac{11 \cdot 1}{11 \cdot 22} - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.2.2.6.2.2

Cancele o fator comum.

Etapa 1.2.2.6.2.3

Reescreva a expressão.

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} | - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.3

Avalie $| \begin{array}{c} \frac{1}{11} & \frac{1}{22} \\ \frac{1}{11} & \frac{1}{22} \end{array} |$ .

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Etapa 1.3.1

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

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} (\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} \cdot \frac{1}{22}) - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.3.2

Simplifique o determinante.

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Etapa 1.3.2.1

Simplifique cada termo.

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Etapa 1.3.2.1.1

Multiplique $\frac{1}{11} \cdot \frac{1}{22}$ .

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Etapa 1.3.2.1.1.1

Multiplique $\frac{1}{11}$ por $\frac{1}{22}$ .

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} (\frac{1}{11 \cdot 22} - \frac{1}{11} \cdot \frac{1}{22}) - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.3.2.1.1.2

Multiplique $11$ por $22$ .

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} (\frac{1}{242} - \frac{1}{11} \cdot \frac{1}{22}) - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.3.2.1.2

Multiplique $- \frac{1}{11} \cdot \frac{1}{22}$ .

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Etapa 1.3.2.1.2.1

Multiplique $\frac{1}{22}$ por $\frac{1}{11}$ .

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} (\frac{1}{242} - \frac{1}{22 \cdot 11}) - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.3.2.1.2.2

Multiplique $22$ por $11$ .

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} (\frac{1}{242} - \frac{1}{242}) - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.3.2.2

Combine os numeradores em relação ao denominador comum.

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} \cdot \frac{1 - 1}{242} - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.3.2.3

Subtraia $1$ de $1$ .

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} \cdot \frac{0}{242} - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.3.2.4

Divida $0$ por $242$ .

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} \cdot 0 - \frac{5}{11} | \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$

Etapa 1.4

Avalie $| \begin{array}{c} \frac{1}{11} & \frac{1}{11} \\ \frac{1}{11} & - \frac{10}{11} \end{array} |$ .

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Etapa 1.4.1

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

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} \cdot 0 - \frac{5}{11} (\frac{1}{11} (- \frac{10}{11}) - \frac{1}{11} \cdot \frac{1}{11})$

Etapa 1.4.2

Simplifique o determinante.

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Etapa 1.4.2.1

Simplifique cada termo.

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Etapa 1.4.2.1.1

Multiplique $\frac{1}{11} (- \frac{10}{11})$ .

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Etapa 1.4.2.1.1.1

Multiplique $\frac{1}{11}$ por $\frac{10}{11}$ .

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} \cdot 0 - \frac{5}{11} (- \frac{10}{11 \cdot 11} - \frac{1}{11} \cdot \frac{1}{11})$

Etapa 1.4.2.1.1.2

Multiplique $11$ por $11$ .

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} \cdot 0 - \frac{5}{11} (- \frac{10}{121} - \frac{1}{11} \cdot \frac{1}{11})$

Etapa 1.4.2.1.2

Multiplique $- \frac{1}{11} \cdot \frac{1}{11}$ .

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Etapa 1.4.2.1.2.1

Multiplique $\frac{1}{11}$ por $\frac{1}{11}$ .

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} \cdot 0 - \frac{5}{11} (- \frac{10}{121} - \frac{1}{11 \cdot 11})$

Etapa 1.4.2.1.2.2

Multiplique $11$ por $11$ .

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} \cdot 0 - \frac{5}{11} (- \frac{10}{121} - \frac{1}{121})$

Etapa 1.4.2.2

Combine os numeradores em relação ao denominador comum.

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} \cdot 0 - \frac{5}{11} \cdot \frac{- 10 - 1}{121}$

Etapa 1.4.2.3

Subtraia $1$ de $- 10$ .

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} \cdot 0 - \frac{5}{11} \cdot \frac{- 11}{121}$

Etapa 1.4.2.4

Cancele o fator comum de $- 11$ e $121$ .

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Etapa 1.4.2.4.1

Fatore $11$ de $- 11$ .

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} \cdot 0 - \frac{5}{11} \cdot \frac{11 (- 1)}{121}$

Etapa 1.4.2.4.2

Cancele os fatores comuns.

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Etapa 1.4.2.4.2.1

Fatore $11$ de $121$ .

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} \cdot 0 - \frac{5}{11} \cdot \frac{11 \cdot - 1}{11 \cdot 11}$

Etapa 1.4.2.4.2.2

Cancele o fator comum.

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} \cdot 0 - \frac{5}{11} \cdot \frac{11 \cdot - 1}{11 \cdot 11}$

Etapa 1.4.2.4.2.3

Reescreva a expressão.

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} \cdot 0 - \frac{5}{11} \cdot \frac{- 1}{11}$

Etapa 1.4.2.5

Mova o número negativo para a frente da fração.

$\frac{1}{11} \cdot \frac{1}{22} - \frac{1}{11} \cdot 0 - \frac{5}{11} (- \frac{1}{11})$

Etapa 1.5

Simplifique o determinante.

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Etapa 1.5.1

Simplifique cada termo.

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Etapa 1.5.1.1

Multiplique $\frac{1}{11} \cdot \frac{1}{22}$ .

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Etapa 1.5.1.1.1

Multiplique $\frac{1}{11}$ por $\frac{1}{22}$ .

$\frac{1}{11 \cdot 22} - \frac{1}{11} \cdot 0 - \frac{5}{11} (- \frac{1}{11})$

Etapa 1.5.1.1.2

Multiplique $11$ por $22$ .

$\frac{1}{242} - \frac{1}{11} \cdot 0 - \frac{5}{11} (- \frac{1}{11})$

Etapa 1.5.1.2

Multiplique $- \frac{1}{11} \cdot 0$ .

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Etapa 1.5.1.2.1

Multiplique $0$ por $- 1$ .

$\frac{1}{242} + 0 (\frac{1}{11}) - \frac{5}{11} (- \frac{1}{11})$

Etapa 1.5.1.2.2

Multiplique $0$ por $\frac{1}{11}$ .

$\frac{1}{242} + 0 - \frac{5}{11} (- \frac{1}{11})$

Etapa 1.5.1.3

Multiplique $- \frac{5}{11} (- \frac{1}{11})$ .

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Etapa 1.5.1.3.1

Multiplique $- 1$ por $- 1$ .

$\frac{1}{242} + 0 + 1 (\frac{5}{11}) \frac{1}{11}$

Etapa 1.5.1.3.2

Multiplique $\frac{5}{11}$ por $1$ .

$\frac{1}{242} + 0 + \frac{5}{11} \cdot \frac{1}{11}$

Etapa 1.5.1.3.3

Multiplique $\frac{5}{11}$ por $\frac{1}{11}$ .

$\frac{1}{242} + 0 + \frac{5}{11 \cdot 11}$

Etapa 1.5.1.3.4

Multiplique $11$ por $11$ .

$\frac{1}{242} + 0 + \frac{5}{121}$

Etapa 1.5.2

Some $\frac{1}{242}$ e $0$ .

$\frac{1}{242} + \frac{5}{121}$

Etapa 1.5.3

Para escrever $\frac{5}{121}$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$\frac{1}{242} + \frac{5}{121} \cdot \frac{2}{2}$

Etapa 1.5.4

Escreva cada expressão com um denominador comum de $242$ , multiplicando cada um por um fator apropriado de $1$ .

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Etapa 1.5.4.1

Multiplique $\frac{5}{121}$ por $\frac{2}{2}$ .

$\frac{1}{242} + \frac{5 \cdot 2}{121 \cdot 2}$

Etapa 1.5.4.2

Multiplique $121$ por $2$ .

$\frac{1}{242} + \frac{5 \cdot 2}{242}$

Etapa 1.5.5

Combine os numeradores em relação ao denominador comum.

$\frac{1 + 5 \cdot 2}{242}$

Etapa 1.5.6

Simplifique o numerador.

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Etapa 1.5.6.1

Multiplique $5$ por $2$ .

$\frac{1 + 10}{242}$

Etapa 1.5.6.2

Some $1$ e $10$ .

$\frac{11}{242}$

Etapa 1.5.7

Cancele o fator comum de $11$ e $242$ .

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Etapa 1.5.7.1

Fatore $11$ de $11$ .

$\frac{11 (1)}{242}$

Etapa 1.5.7.2

Cancele os fatores comuns.

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Etapa 1.5.7.2.1

Fatore $11$ de $242$ .

$\frac{11 \cdot 1}{11 \cdot 22}$

Etapa 1.5.7.2.2

Cancele o fator comum.

$\frac{11 \cdot 1}{11 \cdot 22}$

Etapa 1.5.7.2.3

Reescreva a expressão.

$\frac{1}{22}$

Etapa 2

Since the determinant is non-zero, the inverse exists.

Etapa 3

Set up a $3 \times 6$ matrix where the left half is the original matrix and the right half is its identity matrix.

$[\begin{array}{c} \frac{1}{11} & \frac{1}{11} & - \frac{5}{11} & 1 & 0 & 0 \\ \frac{1}{11} & \frac{1}{11} & \frac{1}{22} & 0 & 1 & 0 \\ \frac{1}{11} & - \frac{10}{11} & \frac{1}{22} & 0 & 0 & 1 \end{array}]$

Etapa 4

Encontre a forma escalonada reduzida por linhas.

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Etapa 4.1

Multiply each element of $R_{1}$ by $11$ to make the entry at $1, 1$ a $1$ .

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Etapa 4.1.1

Multiply each element of $R_{1}$ by $11$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} 11 (\frac{1}{11}) & 11 (\frac{1}{11}) & 11 (- \frac{5}{11}) & 11 \cdot 1 & 11 \cdot 0 & 11 \cdot 0 \\ \frac{1}{11} & \frac{1}{11} & \frac{1}{22} & 0 & 1 & 0 \\ \frac{1}{11} & - \frac{10}{11} & \frac{1}{22} & 0 & 0 & 1 \end{array}]$

Etapa 4.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 1 & - 5 & 11 & 0 & 0 \\ \frac{1}{11} & \frac{1}{11} & \frac{1}{22} & 0 & 1 & 0 \\ \frac{1}{11} & - \frac{10}{11} & \frac{1}{22} & 0 & 0 & 1 \end{array}]$

Etapa 4.2

Perform the row operation $R_{2} = R_{2} - \frac{1}{11} R_{1}$ to make the entry at $2, 1$ a $0$ .

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Etapa 4.2.1

Perform the row operation $R_{2} = R_{2} - \frac{1}{11} R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 5 & 11 & 0 & 0 \\ \frac{1}{11} - \frac{1}{11} \cdot 1 & \frac{1}{11} - \frac{1}{11} \cdot 1 & \frac{1}{22} - \frac{1}{11} \cdot - 5 & 0 - \frac{1}{11} \cdot 11 & 1 - \frac{1}{11} \cdot 0 & 0 - \frac{1}{11} \cdot 0 \\ \frac{1}{11} & - \frac{10}{11} & \frac{1}{22} & 0 & 0 & 1 \end{array}]$

Etapa 4.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 1 & - 5 & 11 & 0 & 0 \\ 0 & 0 & \frac{1}{2} & - 1 & 1 & 0 \\ \frac{1}{11} & - \frac{10}{11} & \frac{1}{22} & 0 & 0 & 1 \end{array}]$

Etapa 4.3

Perform the row operation $R_{3} = R_{3} - \frac{1}{11} R_{1}$ to make the entry at $3, 1$ a $0$ .

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Etapa 4.3.1

Perform the row operation $R_{3} = R_{3} - \frac{1}{11} R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 5 & 11 & 0 & 0 \\ 0 & 0 & \frac{1}{2} & - 1 & 1 & 0 \\ \frac{1}{11} - \frac{1}{11} \cdot 1 & - \frac{10}{11} - \frac{1}{11} \cdot 1 & \frac{1}{22} - \frac{1}{11} \cdot - 5 & 0 - \frac{1}{11} \cdot 11 & 0 - \frac{1}{11} \cdot 0 & 1 - \frac{1}{11} \cdot 0 \end{array}]$

Etapa 4.3.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 5 & 11 & 0 & 0 \\ 0 & 0 & \frac{1}{2} & - 1 & 1 & 0 \\ 0 & - 1 & \frac{1}{2} & - 1 & 0 & 1 \end{array}]$

Etapa 4.4

Swap $R_{3}$ with $R_{2}$ to put a nonzero entry at $2, 2$ .

$[\begin{array}{c} 1 & 1 & - 5 & 11 & 0 & 0 \\ 0 & - 1 & \frac{1}{2} & - 1 & 0 & 1 \\ 0 & 0 & \frac{1}{2} & - 1 & 1 & 0 \end{array}]$

Etapa 4.5

Multiply each element of $R_{2}$ by $- 1$ to make the entry at $2, 2$ a $1$ .

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Etapa 4.5.1

Multiply each element of $R_{2}$ by $- 1$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1 & - 5 & 11 & 0 & 0 \\ - 0 & - - 1 & - \frac{1}{2} & - - 1 & - 0 & - 1 \cdot 1 \\ 0 & 0 & \frac{1}{2} & - 1 & 1 & 0 \end{array}]$

Etapa 4.5.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 1 & - 5 & 11 & 0 & 0 \\ 0 & 1 & - \frac{1}{2} & 1 & 0 & - 1 \\ 0 & 0 & \frac{1}{2} & - 1 & 1 & 0 \end{array}]$

Etapa 4.6

Multiply each element of $R_{3}$ by $2$ to make the entry at $3, 3$ a $1$ .

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Etapa 4.6.1

Multiply each element of $R_{3}$ by $2$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 1 & - 5 & 11 & 0 & 0 \\ 0 & 1 & - \frac{1}{2} & 1 & 0 & - 1 \\ 2 \cdot 0 & 2 \cdot 0 & 2 (\frac{1}{2}) & 2 \cdot - 1 & 2 \cdot 1 & 2 \cdot 0 \end{array}]$

Etapa 4.6.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 5 & 11 & 0 & 0 \\ 0 & 1 & - \frac{1}{2} & 1 & 0 & - 1 \\ 0 & 0 & 1 & - 2 & 2 & 0 \end{array}]$

Etapa 4.7

Perform the row operation $R_{2} = R_{2} + \frac{1}{2} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Etapa 4.7.1

Perform the row operation $R_{2} = R_{2} + \frac{1}{2} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 5 & 11 & 0 & 0 \\ 0 + \frac{1}{2} \cdot 0 & 1 + \frac{1}{2} \cdot 0 & - \frac{1}{2} + \frac{1}{2} \cdot 1 & 1 + \frac{1}{2} \cdot - 2 & 0 + \frac{1}{2} \cdot 2 & - 1 + \frac{1}{2} \cdot 0 \\ 0 & 0 & 1 & - 2 & 2 & 0 \end{array}]$

Etapa 4.7.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 1 & - 5 & 11 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & - 1 \\ 0 & 0 & 1 & - 2 & 2 & 0 \end{array}]$

Etapa 4.8

Perform the row operation $R_{1} = R_{1} + 5 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Etapa 4.8.1

Perform the row operation $R_{1} = R_{1} + 5 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 + 5 \cdot 0 & 1 + 5 \cdot 0 & - 5 + 5 \cdot 1 & 11 + 5 \cdot - 2 & 0 + 5 \cdot 2 & 0 + 5 \cdot 0 \\ 0 & 1 & 0 & 0 & 1 & - 1 \\ 0 & 0 & 1 & - 2 & 2 & 0 \end{array}]$

Etapa 4.8.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 0 & 1 & 10 & 0 \\ 0 & 1 & 0 & 0 & 1 & - 1 \\ 0 & 0 & 1 & - 2 & 2 & 0 \end{array}]$

Etapa 4.9

Perform the row operation $R_{1} = R_{1} - R_{2}$ to make the entry at $1, 2$ a $0$ .

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Etapa 4.9.1

Perform the row operation $R_{1} = R_{1} - R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 0 & 1 - 1 & 0 - 0 & 1 - 0 & 10 - 1 & 0 + 1 \\ 0 & 1 & 0 & 0 & 1 & - 1 \\ 0 & 0 & 1 & - 2 & 2 & 0 \end{array}]$

Etapa 4.9.2

Simplifique $R_{1}$ .

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

Etapa 5

The right half of the reduced row echelon form is the inverse.

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