Álgebra linear Exemplos

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}]$

Etapa 1

Encontre os autovalores.

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Etapa 1.1

Estabeleça a fórmula para encontrar a equação característica $p (λ)$ .

$p (λ) = determinante (A - λ I_{3})$

Etapa 1.2

A matriz identidade ou matriz unitária de tamanho $3$ é a matriz quadrada $3 \times 3$ com números "um" na diagonal principal e zeros nos outros lugares.

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

Etapa 1.3

Substitua os valores conhecidos em $p (λ) = determinante (A - λ I_{3})$ .

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Etapa 1.3.1

Substitua $[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}]$ por $A$ .

$p (λ) = determinante ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] - λ I_{3})$

Etapa 1.3.2

Substitua $[\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$ por $I_{3}$ .

$p (λ) = determinante ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Etapa 1.4

Simplifique.

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Etapa 1.4.1

Simplifique cada termo.

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Etapa 1.4.1.1

Multiplique $- λ$ por cada elemento da matriz.

$p (λ) = determinante ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2

Simplifique cada elemento da matriz.

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Etapa 1.4.1.2.1

Multiplique $- 1$ por $1$ .

$p (λ) = determinante ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.2

Multiplique $- λ \cdot 0$ .

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Etapa 1.4.1.2.2.1

Multiplique $0$ por $- 1$ .

$p (λ) = determinante ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - λ & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.2.2

Multiplique $0$ por $λ$ .

$p (λ) = determinante ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.3

Multiplique $- λ \cdot 0$ .

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Etapa 1.4.1.2.3.1

Multiplique $0$ por $- 1$ .

$p (λ) = determinante ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - λ & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.3.2

Multiplique $0$ por $λ$ .

$p (λ) = determinante ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.4

Multiplique $- λ \cdot 0$ .

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Etapa 1.4.1.2.4.1

Multiplique $0$ por $- 1$ .

$p (λ) = determinante ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 λ & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.4.2

Multiplique $0$ por $λ$ .

$p (λ) = determinante ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.5

Multiplique $- 1$ por $1$ .

$p (λ) = determinante ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.6

Multiplique $- λ \cdot 0$ .

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Etapa 1.4.1.2.6.1

Multiplique $0$ por $- 1$ .

$p (λ) = determinante ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.6.2

Multiplique $0$ por $λ$ .

$p (λ) = determinante ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.7

Multiplique $- λ \cdot 0$ .

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Etapa 1.4.1.2.7.1

Multiplique $0$ por $- 1$ .

$p (λ) = determinante ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.7.2

Multiplique $0$ por $λ$ .

$p (λ) = determinante ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.8

Multiplique $- λ \cdot 0$ .

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Etapa 1.4.1.2.8.1

Multiplique $0$ por $- 1$ .

$p (λ) = determinante ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 λ & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.8.2

Multiplique $0$ por $λ$ .

$p (λ) = determinante ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.9

Multiplique $- 1$ por $1$ .

$p (λ) = determinante ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \end{array}])$

Etapa 1.4.2

Adicione os elementos correspondentes.

$p (λ) = determinante [\begin{array}{c} 0.8 - λ & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - λ & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - λ \end{array}]$

Etapa 1.4.3

Simplify each element.

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Etapa 1.4.3.1

Some $0.2$ e $0$ .

$p (λ) = determinante [\begin{array}{c} 0.8 - λ & 0.2 & 0.2 + 0 \\ 0 + 0 & 0.5 - λ & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - λ \end{array}]$

Etapa 1.4.3.2

Some $0.2$ e $0$ .

$p (λ) = determinante [\begin{array}{c} 0.8 - λ & 0.2 & 0.2 \\ 0 + 0 & 0.5 - λ & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - λ \end{array}]$

Etapa 1.4.3.3

Some $0$ e $0$ .

$p (λ) = determinante [\begin{array}{c} 0.8 - λ & 0.2 & 0.2 \\ 0 & 0.5 - λ & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - λ \end{array}]$

Etapa 1.4.3.4

Some $0.1$ e $0$ .

$p (λ) = determinante [\begin{array}{c} 0.8 - λ & 0.2 & 0.2 \\ 0 & 0.5 - λ & 0.1 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - λ \end{array}]$

Etapa 1.4.3.5

Some $0.2$ e $0$ .

$p (λ) = determinante [\begin{array}{c} 0.8 - λ & 0.2 & 0.2 \\ 0 & 0.5 - λ & 0.1 \\ 0.2 & 0.3 + 0 & 0.7 - λ \end{array}]$

Etapa 1.4.3.6

Some $0.3$ e $0$ .

$p (λ) = determinante [\begin{array}{c} 0.8 - λ & 0.2 & 0.2 \\ 0 & 0.5 - λ & 0.1 \\ 0.2 & 0.3 & 0.7 - λ \end{array}]$

Etapa 1.5

Find the determinant.

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Etapa 1.5.1

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 1.5.1.1

Consider the corresponding sign chart.

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

Etapa 1.5.1.2

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

Etapa 1.5.1.3

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

$| \begin{array}{c} 0.5 - λ & 0.1 \\ 0.3 & 0.7 - λ \end{array} |$

Etapa 1.5.1.4

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

$(0.8 - λ) | \begin{array}{c} 0.5 - λ & 0.1 \\ 0.3 & 0.7 - λ \end{array} |$

Etapa 1.5.1.5

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

$| \begin{array}{c} 0.2 & 0.2 \\ 0.3 & 0.7 - λ \end{array} |$

Etapa 1.5.1.6

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

$0 | \begin{array}{c} 0.2 & 0.2 \\ 0.3 & 0.7 - λ \end{array} |$

Etapa 1.5.1.7

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

$| \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$

Etapa 1.5.1.8

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

$0.2 | \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$

Etapa 1.5.1.9

Add the terms together.

$p (λ) = (0.8 - λ) | \begin{array}{c} 0.5 - λ & 0.1 \\ 0.3 & 0.7 - λ \end{array} | + 0 | \begin{array}{c} 0.2 & 0.2 \\ 0.3 & 0.7 - λ \end{array} | + 0.2 | \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$

Etapa 1.5.2

Multiplique $0$ por $| \begin{array}{c} 0.2 & 0.2 \\ 0.3 & 0.7 - λ \end{array} |$ .

$p (λ) = (0.8 - λ) | \begin{array}{c} 0.5 - λ & 0.1 \\ 0.3 & 0.7 - λ \end{array} | + 0 + 0.2 | \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$

Etapa 1.5.3

Avalie $| \begin{array}{c} 0.5 - λ & 0.1 \\ 0.3 & 0.7 - λ \end{array} |$ .

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Etapa 1.5.3.1

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

$p (λ) = (0.8 - λ) ((0.5 - λ) (0.7 - λ) - 0.3 \cdot 0.1) + 0 + 0.2 | \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$

Etapa 1.5.3.2

Simplifique o determinante.

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Etapa 1.5.3.2.1

Simplifique cada termo.

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Etapa 1.5.3.2.1.1

Expanda $(0.5 - λ) (0.7 - λ)$ usando o método FOIL.

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Etapa 1.5.3.2.1.1.1

Aplique a propriedade distributiva.

$p (λ) = (0.8 - λ) (0.5 (0.7 - λ) - λ (0.7 - λ) - 0.3 \cdot 0.1) + 0 + 0.2 | \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$

Etapa 1.5.3.2.1.1.2

Aplique a propriedade distributiva.

$p (λ) = (0.8 - λ) (0.5 \cdot 0.7 + 0.5 (- λ) - λ (0.7 - λ) - 0.3 \cdot 0.1) + 0 + 0.2 | \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$

Etapa 1.5.3.2.1.1.3

Aplique a propriedade distributiva.

$p (λ) = (0.8 - λ) (0.5 \cdot 0.7 + 0.5 (- λ) - λ \cdot 0.7 - λ (- λ) - 0.3 \cdot 0.1) + 0 + 0.2 | \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$

Etapa 1.5.3.2.1.2

Simplifique e combine termos semelhantes.

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Etapa 1.5.3.2.1.2.1

Simplifique cada termo.

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Etapa 1.5.3.2.1.2.1.1

Multiplique $0.5$ por $0.7$ .

$p (λ) = (0.8 - λ) (0.35 + 0.5 (- λ) - λ \cdot 0.7 - λ (- λ) - 0.3 \cdot 0.1) + 0 + 0.2 | \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$

Etapa 1.5.3.2.1.2.1.2

Multiplique $- 1$ por $0.5$ .

$p (λ) = (0.8 - λ) (0.35 - 0.5 λ - λ \cdot 0.7 - λ (- λ) - 0.3 \cdot 0.1) + 0 + 0.2 | \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$

Etapa 1.5.3.2.1.2.1.3

Multiplique $0.7$ por $- 1$ .

$p (λ) = (0.8 - λ) (0.35 - 0.5 λ - 0.7 λ - λ (- λ) - 0.3 \cdot 0.1) + 0 + 0.2 | \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$

Etapa 1.5.3.2.1.2.1.4

Reescreva usando a propriedade comutativa da multiplicação.

$p (λ) = (0.8 - λ) (0.35 - 0.5 λ - 0.7 λ - 1 \cdot - 1 λ \cdot λ - 0.3 \cdot 0.1) + 0 + 0.2 | \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$

Etapa 1.5.3.2.1.2.1.5

Multiplique $λ$ por $λ$ somando os expoentes.

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Etapa 1.5.3.2.1.2.1.5.1

Mova $λ$ .

$p (λ) = (0.8 - λ) (0.35 - 0.5 λ - 0.7 λ - 1 \cdot - 1 (λ \cdot λ) - 0.3 \cdot 0.1) + 0 + 0.2 | \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$

Etapa 1.5.3.2.1.2.1.5.2

Multiplique $λ$ por $λ$ .

$p (λ) = (0.8 - λ) (0.35 - 0.5 λ - 0.7 λ - 1 \cdot - 1 λ^{2} - 0.3 \cdot 0.1) + 0 + 0.2 | \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$

Etapa 1.5.3.2.1.2.1.6

Multiplique $- 1$ por $- 1$ .

$p (λ) = (0.8 - λ) (0.35 - 0.5 λ - 0.7 λ + 1 λ^{2} - 0.3 \cdot 0.1) + 0 + 0.2 | \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$

Etapa 1.5.3.2.1.2.1.7

Multiplique $λ^{2}$ por $1$ .

$p (λ) = (0.8 - λ) (0.35 - 0.5 λ - 0.7 λ + λ^{2} - 0.3 \cdot 0.1) + 0 + 0.2 | \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$

Etapa 1.5.3.2.1.2.2

Subtraia $0.7 λ$ de $- 0.5 λ$ .

$p (λ) = (0.8 - λ) (0.35 - 1.2 λ + λ^{2} - 0.3 \cdot 0.1) + 0 + 0.2 | \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$

Etapa 1.5.3.2.1.3

Multiplique $- 0.3$ por $0.1$ .

$p (λ) = (0.8 - λ) (0.35 - 1.2 λ + λ^{2} - 0.03) + 0 + 0.2 | \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$

Etapa 1.5.3.2.2

Subtraia $0.03$ de $0.35$ .

$p (λ) = (0.8 - λ) (- 1.2 λ + λ^{2} + 0.32) + 0 + 0.2 | \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$

Etapa 1.5.3.2.3

Reordene $- 1.2 λ$ e $λ^{2}$ .

$p (λ) = (0.8 - λ) (λ^{2} - 1.2 λ + 0.32) + 0 + 0.2 | \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$

Etapa 1.5.4

Avalie $| \begin{array}{c} 0.2 & 0.2 \\ 0.5 - λ & 0.1 \end{array} |$ .

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Etapa 1.5.4.1

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

$p (λ) = (0.8 - λ) (λ^{2} - 1.2 λ + 0.32) + 0 + 0.2 (0.2 \cdot 0.1 - (0.5 - λ) \cdot 0.2)$

Etapa 1.5.4.2

Simplifique o determinante.

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Etapa 1.5.4.2.1

Simplifique cada termo.

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Etapa 1.5.4.2.1.1

Multiplique $0.2$ por $0.1$ .

$p (λ) = (0.8 - λ) (λ^{2} - 1.2 λ + 0.32) + 0 + 0.2 (0.02 - (0.5 - λ) \cdot 0.2)$

Etapa 1.5.4.2.1.2

Aplique a propriedade distributiva.

$p (λ) = (0.8 - λ) (λ^{2} - 1.2 λ + 0.32) + 0 + 0.2 (0.02 + (- 1 \cdot 0.5 - - λ) \cdot 0.2)$

Etapa 1.5.4.2.1.3

Multiplique $- 1$ por $0.5$ .

$p (λ) = (0.8 - λ) (λ^{2} - 1.2 λ + 0.32) + 0 + 0.2 (0.02 + (- 0.5 - - λ) \cdot 0.2)$

Etapa 1.5.4.2.1.4

Multiplique $- - λ$ .

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Etapa 1.5.4.2.1.4.1

Multiplique $- 1$ por $- 1$ .

$p (λ) = (0.8 - λ) (λ^{2} - 1.2 λ + 0.32) + 0 + 0.2 (0.02 + (- 0.5 + 1 λ) \cdot 0.2)$

Etapa 1.5.4.2.1.4.2

Multiplique $λ$ por $1$ .

$p (λ) = (0.8 - λ) (λ^{2} - 1.2 λ + 0.32) + 0 + 0.2 (0.02 + (- 0.5 + λ) \cdot 0.2)$

Etapa 1.5.4.2.1.5

Aplique a propriedade distributiva.

$p (λ) = (0.8 - λ) (λ^{2} - 1.2 λ + 0.32) + 0 + 0.2 (0.02 - 0.5 \cdot 0.2 + λ \cdot 0.2)$

Etapa 1.5.4.2.1.6

Multiplique $- 0.5$ por $0.2$ .

$p (λ) = (0.8 - λ) (λ^{2} - 1.2 λ + 0.32) + 0 + 0.2 (0.02 - 0.1 + λ \cdot 0.2)$

Etapa 1.5.4.2.1.7

Mova $0.2$ para a esquerda de $λ$ .

$p (λ) = (0.8 - λ) (λ^{2} - 1.2 λ + 0.32) + 0 + 0.2 (0.02 - 0.1 + 0.2 λ)$

Etapa 1.5.4.2.2

Subtraia $0.1$ de $0.02$ .

$p (λ) = (0.8 - λ) (λ^{2} - 1.2 λ + 0.32) + 0 + 0.2 (- 0.08 + 0.2 λ)$

Etapa 1.5.4.2.3

Reordene $- 0.08$ e $0.2 λ$ .

$p (λ) = (0.8 - λ) (λ^{2} - 1.2 λ + 0.32) + 0 + 0.2 (0.2 λ - 0.08)$

Etapa 1.5.5

Simplifique o determinante.

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Etapa 1.5.5.1

Some $(0.8 - λ) (λ^{2} - 1.2 λ + 0.32)$ e $0$ .

$p (λ) = (0.8 - λ) (λ^{2} - 1.2 λ + 0.32) + 0.2 (0.2 λ - 0.08)$

Etapa 1.5.5.2

Simplifique cada termo.

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Etapa 1.5.5.2.1

Expanda $(0.8 - λ) (λ^{2} - 1.2 λ + 0.32)$ multiplicando cada termo na primeira expressão por cada um dos termos na segunda expressão.

$p (λ) = 0.8 λ^{2} + 0.8 (- 1.2 λ) + 0.8 \cdot 0.32 - λ \cdot λ^{2} - λ (- 1.2 λ) - λ \cdot 0.32 + 0.2 (0.2 λ - 0.08)$

Etapa 1.5.5.2.2

Simplifique cada termo.

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Etapa 1.5.5.2.2.1

Multiplique $- 1.2$ por $0.8$ .

$p (λ) = 0.8 λ^{2} - 0.96 λ + 0.8 \cdot 0.32 - λ \cdot λ^{2} - λ (- 1.2 λ) - λ \cdot 0.32 + 0.2 (0.2 λ - 0.08)$

Etapa 1.5.5.2.2.2

Multiplique $0.8$ por $0.32$ .

$p (λ) = 0.8 λ^{2} - 0.96 λ + 0.256 - λ \cdot λ^{2} - λ (- 1.2 λ) - λ \cdot 0.32 + 0.2 (0.2 λ - 0.08)$

Etapa 1.5.5.2.2.3

Multiplique $λ$ por $λ^{2}$ somando os expoentes.

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Etapa 1.5.5.2.2.3.1

Mova $λ^{2}$ .

$p (λ) = 0.8 λ^{2} - 0.96 λ + 0.256 - (λ^{2} λ) - λ (- 1.2 λ) - λ \cdot 0.32 + 0.2 (0.2 λ - 0.08)$

Etapa 1.5.5.2.2.3.2

Multiplique $λ^{2}$ por $λ$ .

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Etapa 1.5.5.2.2.3.2.1

Eleve $λ$ à potência de $1$ .

$p (λ) = 0.8 λ^{2} - 0.96 λ + 0.256 - (λ^{2} λ^{1}) - λ (- 1.2 λ) - λ \cdot 0.32 + 0.2 (0.2 λ - 0.08)$

Etapa 1.5.5.2.2.3.2.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$p (λ) = 0.8 λ^{2} - 0.96 λ + 0.256 - λ^{2 + 1} - λ (- 1.2 λ) - λ \cdot 0.32 + 0.2 (0.2 λ - 0.08)$

Etapa 1.5.5.2.2.3.3

Some $2$ e $1$ .

$p (λ) = 0.8 λ^{2} - 0.96 λ + 0.256 - λ^{3} - λ (- 1.2 λ) - λ \cdot 0.32 + 0.2 (0.2 λ - 0.08)$

Etapa 1.5.5.2.2.4

Reescreva usando a propriedade comutativa da multiplicação.

$p (λ) = 0.8 λ^{2} - 0.96 λ + 0.256 - λ^{3} - 1 \cdot - 1.2 λ \cdot λ - λ \cdot 0.32 + 0.2 (0.2 λ - 0.08)$

Etapa 1.5.5.2.2.5

Multiplique $λ$ por $λ$ somando os expoentes.

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Etapa 1.5.5.2.2.5.1

Mova $λ$ .

$p (λ) = 0.8 λ^{2} - 0.96 λ + 0.256 - λ^{3} - 1 \cdot - 1.2 (λ \cdot λ) - λ \cdot 0.32 + 0.2 (0.2 λ - 0.08)$

Etapa 1.5.5.2.2.5.2

Multiplique $λ$ por $λ$ .

$p (λ) = 0.8 λ^{2} - 0.96 λ + 0.256 - λ^{3} - 1 \cdot - 1.2 λ^{2} - λ \cdot 0.32 + 0.2 (0.2 λ - 0.08)$

Etapa 1.5.5.2.2.6

Multiplique $- 1$ por $- 1.2$ .

$p (λ) = 0.8 λ^{2} - 0.96 λ + 0.256 - λ^{3} + 1.2 λ^{2} - λ \cdot 0.32 + 0.2 (0.2 λ - 0.08)$

Etapa 1.5.5.2.2.7

Multiplique $0.32$ por $- 1$ .

$p (λ) = 0.8 λ^{2} - 0.96 λ + 0.256 - λ^{3} + 1.2 λ^{2} - 0.32 λ + 0.2 (0.2 λ - 0.08)$

Etapa 1.5.5.2.3

Some $0.8 λ^{2}$ e $1.2 λ^{2}$ .

$p (λ) = 2 λ^{2} - 0.96 λ + 0.256 - λ^{3} - 0.32 λ + 0.2 (0.2 λ - 0.08)$

Etapa 1.5.5.2.4

Subtraia $0.32 λ$ de $- 0.96 λ$ .

$p (λ) = 2 λ^{2} - 1.28 λ + 0.256 - λ^{3} + 0.2 (0.2 λ - 0.08)$

Etapa 1.5.5.2.5

Aplique a propriedade distributiva.

$p (λ) = 2 λ^{2} - 1.28 λ + 0.256 - λ^{3} + 0.2 (0.2 λ) + 0.2 \cdot - 0.08$

Etapa 1.5.5.2.6

Multiplique $0.2$ por $0.2$ .

$p (λ) = 2 λ^{2} - 1.28 λ + 0.256 - λ^{3} + 0.04 λ + 0.2 \cdot - 0.08$

Etapa 1.5.5.2.7

Multiplique $0.2$ por $- 0.08$ .

$p (λ) = 2 λ^{2} - 1.28 λ + 0.256 - λ^{3} + 0.04 λ - 0.016$

Etapa 1.5.5.3

Some $- 1.28 λ$ e $0.04 λ$ .

$p (λ) = 2 λ^{2} - 1.24 λ + 0.256 - λ^{3} - 0.016$

Etapa 1.5.5.4

Subtraia $0.016$ de $0.256$ .

$p (λ) = 2 λ^{2} - 1.24 λ - λ^{3} + 0.24$

Etapa 1.5.5.5

Mova $- 1.24 λ$ .

$p (λ) = 2 λ^{2} - λ^{3} - 1.24 λ + 0.24$

Etapa 1.5.5.6

Reordene $2 λ^{2}$ e $- λ^{3}$ .

$p (λ) = - λ^{3} + 2 λ^{2} - 1.24 λ + 0.24$

Etapa 1.6

Defina o polinômio característico como igual a $0$ para encontrar os autovalores $λ$ .

$- λ^{3} + 2 λ^{2} - 1.24 λ + 0.24 = 0$

Etapa 1.7

Resolva $λ$ .

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Etapa 1.7.1

Represente cada lado da equação em um gráfico. A solução é o valor x do ponto de intersecção.

$λ = \frac{2}{5}, \frac{3}{5}, 1$

Etapa 2

The eigenvector is equal to the null space of the matrix minus the eigenvalue times the identity matrix where $N$ is the null space and $I$ is the identity matrix.

$ε_{A} = N (A - λ I_{3})$

Etapa 3

Find the eigenvector using the eigenvalue $λ = \frac{2}{5}$ .

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Etapa 3.1

Substitua os valores conhecidos na fórmula.

$N ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] - \frac{2}{5} [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Etapa 3.2

Simplifique.

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Etapa 3.2.1

Simplifique cada termo.

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Etapa 3.2.1.1

Multiplique $- \frac{2}{5}$ por cada elemento da matriz.

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{2}{5} \cdot 1 & - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 0 \\ - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 1 & - \frac{2}{5} \cdot 0 \\ - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 1 \end{array}]$

Etapa 3.2.1.2

Simplifique cada elemento da matriz.

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Etapa 3.2.1.2.1

Multiplique $- 1$ por $1$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{2}{5} & - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 0 \\ - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 1 & - \frac{2}{5} \cdot 0 \\ - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 1 \end{array}]$

Etapa 3.2.1.2.2

Multiplique $- \frac{2}{5} \cdot 0$ .

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Etapa 3.2.1.2.2.1

Multiplique $0$ por $- 1$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{2}{5} & 0 (\frac{2}{5}) & - \frac{2}{5} \cdot 0 \\ - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 1 & - \frac{2}{5} \cdot 0 \\ - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 1 \end{array}]$

Etapa 3.2.1.2.2.2

Multiplique $0$ por $\frac{2}{5}$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{2}{5} & 0 & - \frac{2}{5} \cdot 0 \\ - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 1 & - \frac{2}{5} \cdot 0 \\ - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 1 \end{array}]$

Etapa 3.2.1.2.3

Multiplique $- \frac{2}{5} \cdot 0$ .

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Etapa 3.2.1.2.3.1

Multiplique $0$ por $- 1$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{2}{5} & 0 & 0 (\frac{2}{5}) \\ - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 1 & - \frac{2}{5} \cdot 0 \\ - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 1 \end{array}]$

Etapa 3.2.1.2.3.2

Multiplique $0$ por $\frac{2}{5}$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{2}{5} & 0 & 0 \\ - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 1 & - \frac{2}{5} \cdot 0 \\ - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 1 \end{array}]$

Etapa 3.2.1.2.4

Multiplique $- \frac{2}{5} \cdot 0$ .

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Etapa 3.2.1.2.4.1

Multiplique $0$ por $- 1$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{2}{5} & 0 & 0 \\ 0 (\frac{2}{5}) & - \frac{2}{5} \cdot 1 & - \frac{2}{5} \cdot 0 \\ - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 1 \end{array}]$

Etapa 3.2.1.2.4.2

Multiplique $0$ por $\frac{2}{5}$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{2}{5} & 0 & 0 \\ 0 & - \frac{2}{5} \cdot 1 & - \frac{2}{5} \cdot 0 \\ - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 1 \end{array}]$

Etapa 3.2.1.2.5

Multiplique $- 1$ por $1$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{2}{5} & 0 & 0 \\ 0 & - \frac{2}{5} & - \frac{2}{5} \cdot 0 \\ - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 1 \end{array}]$

Etapa 3.2.1.2.6

Multiplique $- \frac{2}{5} \cdot 0$ .

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Etapa 3.2.1.2.6.1

Multiplique $0$ por $- 1$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{2}{5} & 0 & 0 \\ 0 & - \frac{2}{5} & 0 (\frac{2}{5}) \\ - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 1 \end{array}]$

Etapa 3.2.1.2.6.2

Multiplique $0$ por $\frac{2}{5}$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{2}{5} & 0 & 0 \\ 0 & - \frac{2}{5} & 0 \\ - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 1 \end{array}]$

Etapa 3.2.1.2.7

Multiplique $- \frac{2}{5} \cdot 0$ .

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Etapa 3.2.1.2.7.1

Multiplique $0$ por $- 1$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{2}{5} & 0 & 0 \\ 0 & - \frac{2}{5} & 0 \\ 0 (\frac{2}{5}) & - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 1 \end{array}]$

Etapa 3.2.1.2.7.2

Multiplique $0$ por $\frac{2}{5}$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{2}{5} & 0 & 0 \\ 0 & - \frac{2}{5} & 0 \\ 0 & - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 1 \end{array}]$

Etapa 3.2.1.2.8

Multiplique $- \frac{2}{5} \cdot 0$ .

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Etapa 3.2.1.2.8.1

Multiplique $0$ por $- 1$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{2}{5} & 0 & 0 \\ 0 & - \frac{2}{5} & 0 \\ 0 & 0 (\frac{2}{5}) & - \frac{2}{5} \cdot 1 \end{array}]$

Etapa 3.2.1.2.8.2

Multiplique $0$ por $\frac{2}{5}$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{2}{5} & 0 & 0 \\ 0 & - \frac{2}{5} & 0 \\ 0 & 0 & - \frac{2}{5} \cdot 1 \end{array}]$

Etapa 3.2.1.2.9

Multiplique $- 1$ por $1$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{2}{5} & 0 & 0 \\ 0 & - \frac{2}{5} & 0 \\ 0 & 0 & - \frac{2}{5} \end{array}]$

Etapa 3.2.2

Adicione os elementos correspondentes.

$[\begin{array}{c} 0.8 - \frac{2}{5} & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{2}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3

Simplify each element.

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Etapa 3.2.3.1

Para escrever $0.8$ como fração com um denominador comum, multiplique por $\frac{5}{5}$ .

$[\begin{array}{c} 0.8 \cdot \frac{5}{5} - \frac{2}{5} & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{2}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.2

Combine $0.8$ e $\frac{5}{5}$ .

$[\begin{array}{c} \frac{0.8 \cdot 5}{5} - \frac{2}{5} & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{2}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.3

Combine os numeradores em relação ao denominador comum.

$[\begin{array}{c} \frac{0.8 \cdot 5 - 2}{5} & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{2}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.4

Simplifique o numerador.

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Etapa 3.2.3.4.1

Multiplique $0.8$ por $5$ .

$[\begin{array}{c} \frac{4 - 2}{5} & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{2}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.4.2

Subtraia $2$ de $4$ .

$[\begin{array}{c} \frac{2}{5} & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{2}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.5

Cancele o fator comum de $2$ e $5$ .

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Etapa 3.2.3.5.1

Reescreva $2$ como $1 (2)$ .

$[\begin{array}{c} \frac{1 (2)}{5} & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{2}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.5.2

Cancele os fatores comuns.

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Etapa 3.2.3.5.2.1

Reescreva $5$ como $1 (5)$ .

$[\begin{array}{c} \frac{1 \cdot 2}{1 \cdot 5} & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{2}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.5.2.2

Cancele o fator comum.

$[\begin{array}{c} \frac{1 \cdot 2}{1 \cdot 5} & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{2}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.5.2.3

Reescreva a expressão.

$[\begin{array}{c} \frac{2}{5} & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{2}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.6

Some $0.2$ e $0$ .

$[\begin{array}{c} \frac{2}{5} & 0.2 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{2}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.7

Some $0.2$ e $0$ .

$[\begin{array}{c} \frac{2}{5} & 0.2 & 0.2 \\ 0 + 0 & 0.5 - \frac{2}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.8

Some $0$ e $0$ .

$[\begin{array}{c} \frac{2}{5} & 0.2 & 0.2 \\ 0 & 0.5 - \frac{2}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.9

Para escrever $0.5$ como fração com um denominador comum, multiplique por $\frac{5}{5}$ .

$[\begin{array}{c} \frac{2}{5} & 0.2 & 0.2 \\ 0 & 0.5 \cdot \frac{5}{5} - \frac{2}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.10

Combine $0.5$ e $\frac{5}{5}$ .

$[\begin{array}{c} \frac{2}{5} & 0.2 & 0.2 \\ 0 & \frac{0.5 \cdot 5}{5} - \frac{2}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.11

Combine os numeradores em relação ao denominador comum.

$[\begin{array}{c} \frac{2}{5} & 0.2 & 0.2 \\ 0 & \frac{0.5 \cdot 5 - 2}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.12

Simplifique o numerador.

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Etapa 3.2.3.12.1

Multiplique $0.5$ por $5$ .

$[\begin{array}{c} \frac{2}{5} & 0.2 & 0.2 \\ 0 & \frac{2.5 - 2}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.12.2

Subtraia $2$ de $2.5$ .

$[\begin{array}{c} \frac{2}{5} & 0.2 & 0.2 \\ 0 & \frac{0.5}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.13

Divida $0.5$ por $5$ .

$[\begin{array}{c} \frac{2}{5} & 0.2 & 0.2 \\ 0 & 0.1 & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.14

Some $0.1$ e $0$ .

$[\begin{array}{c} \frac{2}{5} & 0.2 & 0.2 \\ 0 & 0.1 & 0.1 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.15

Some $0.2$ e $0$ .

$[\begin{array}{c} \frac{2}{5} & 0.2 & 0.2 \\ 0 & 0.1 & 0.1 \\ 0.2 & 0.3 + 0 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.16

Some $0.3$ e $0$ .

$[\begin{array}{c} \frac{2}{5} & 0.2 & 0.2 \\ 0 & 0.1 & 0.1 \\ 0.2 & 0.3 & 0.7 - \frac{2}{5} \end{array}]$

Etapa 3.2.3.17

Para escrever $0.7$ como fração com um denominador comum, multiplique por $\frac{5}{5}$ .

$[\begin{array}{c} \frac{2}{5} & 0.2 & 0.2 \\ 0 & 0.1 & 0.1 \\ 0.2 & 0.3 & 0.7 \cdot \frac{5}{5} - \frac{2}{5} \end{array}]$

Etapa 3.2.3.18

Combine $0.7$ e $\frac{5}{5}$ .

$[\begin{array}{c} \frac{2}{5} & 0.2 & 0.2 \\ 0 & 0.1 & 0.1 \\ 0.2 & 0.3 & \frac{0.7 \cdot 5}{5} - \frac{2}{5} \end{array}]$

Etapa 3.2.3.19

Combine os numeradores em relação ao denominador comum.

$[\begin{array}{c} \frac{2}{5} & 0.2 & 0.2 \\ 0 & 0.1 & 0.1 \\ 0.2 & 0.3 & \frac{0.7 \cdot 5 - 2}{5} \end{array}]$

Etapa 3.2.3.20

Simplifique o numerador.

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Etapa 3.2.3.20.1

Multiplique $0.7$ por $5$ .

$[\begin{array}{c} \frac{2}{5} & 0.2 & 0.2 \\ 0 & 0.1 & 0.1 \\ 0.2 & 0.3 & \frac{3.5 - 2}{5} \end{array}]$

Etapa 3.2.3.20.2

Subtraia $2$ de $3.5$ .

$[\begin{array}{c} \frac{2}{5} & 0.2 & 0.2 \\ 0 & 0.1 & 0.1 \\ 0.2 & 0.3 & \frac{1.5}{5} \end{array}]$

Etapa 3.2.3.21

Divida $1.5$ por $5$ .

$[\begin{array}{c} \frac{2}{5} & 0.2 & 0.2 \\ 0 & 0.1 & 0.1 \\ 0.2 & 0.3 & 0.3 \end{array}]$

Etapa 3.3

Find the null space when $λ = \frac{2}{5}$ .

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Etapa 3.3.1

Write as an augmented matrix for $A x = 0$ .

$[\begin{array}{c} \frac{2}{5} & 0.2 & 0.2 & 0 \\ 0 & 0.1 & 0.1 & 0 \\ 0.2 & 0.3 & 0.3 & 0 \end{array}]$

Etapa 3.3.2

Encontre a forma escalonada reduzida por linhas.

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Etapa 3.3.2.1

Multiply each element of $R_{1}$ by $\frac{5}{2}$ to make the entry at $1, 1$ a $1$ .

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Etapa 3.3.2.1.1

Multiply each element of $R_{1}$ by $\frac{5}{2}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{5}{2} \cdot \frac{2}{5} & \frac{5}{2} \cdot 0.2 & \frac{5}{2} \cdot 0.2 & \frac{5}{2} \cdot 0 \\ 0 & 0.1 & 0.1 & 0 \\ 0.2 & 0.3 & 0.3 & 0 \end{array}]$

Etapa 3.3.2.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & 0.1 & 0.1 & 0 \\ 0.2 & 0.3 & 0.3 & 0 \end{array}]$

Etapa 3.3.2.2

Perform the row operation $R_{3} = R_{3} - 0.2 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Etapa 3.3.2.2.1

Perform the row operation $R_{3} = R_{3} - 0.2 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & 0.1 & 0.1 & 0 \\ 0.2 - 0.2 \cdot 1 & 0.3 - 0.2 (\frac{1}{2}) & 0.3 - 0.2 (\frac{1}{2}) & 0 - 0.2 \cdot 0 \end{array}]$

Etapa 3.3.2.2.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & 0.1 & 0.1 & 0 \\ 0 & \frac{1}{5} & \frac{1}{5} & 0 \end{array}]$

Etapa 3.3.2.3

Multiply each element of $R_{2}$ by $\frac{1}{0.1}$ to make the entry at $2, 2$ a $1$ .

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Etapa 3.3.2.3.1

Multiply each element of $R_{2}$ by $\frac{1}{0.1}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & 0 \\ \frac{0}{0.1} & \frac{0.1}{0.1} & \frac{0.1}{0.1} & \frac{0}{0.1} \\ 0 & \frac{1}{5} & \frac{1}{5} & 0 \end{array}]$

Etapa 3.3.2.3.2

Simplifique $R_{2}$ .

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

Etapa 3.3.2.4

Perform the row operation $R_{3} = R_{3} - \frac{1}{5} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Etapa 3.3.2.4.1

Perform the row operation $R_{3} = R_{3} - \frac{1}{5} R_{2}$ to make the entry at $3, 2$ a $0$ .

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

Etapa 3.3.2.4.2

Simplifique $R_{3}$ .

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

Etapa 3.3.2.5

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

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Etapa 3.3.2.5.1

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

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

Etapa 3.3.2.5.2

Simplifique $R_{1}$ .

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

Etapa 3.3.3

Use the result matrix to declare the final solution to the system of equations.

$x = 0$

$y + z = 0$

$0 = 0$

Etapa 3.3.4

Write a solution vector by solving in terms of the free variables in each row.

$[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ - z \\ z \end{array}]$

Etapa 3.3.5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x \\ y \\ z \end{array}] = z [\begin{array}{c} 0 \\ - 1 \\ 1 \end{array}]$

Etapa 3.3.6

Write as a solution set.

${z [\begin{array}{c} 0 \\ - 1 \\ 1 \end{array}] | z \in R}$

Etapa 3.3.7

The solution is the set of vectors created from the free variables of the system.

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

Etapa 4

Find the eigenvector using the eigenvalue $λ = \frac{3}{5}$ .

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Etapa 4.1

Substitua os valores conhecidos na fórmula.

$N ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] - \frac{3}{5} [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Etapa 4.2

Simplifique.

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Etapa 4.2.1

Simplifique cada termo.

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Etapa 4.2.1.1

Multiplique $- \frac{3}{5}$ por cada elemento da matriz.

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{3}{5} \cdot 1 & - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 0 \\ - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 1 & - \frac{3}{5} \cdot 0 \\ - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 1 \end{array}]$

Etapa 4.2.1.2

Simplifique cada elemento da matriz.

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Etapa 4.2.1.2.1

Multiplique $- 1$ por $1$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{3}{5} & - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 0 \\ - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 1 & - \frac{3}{5} \cdot 0 \\ - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 1 \end{array}]$

Etapa 4.2.1.2.2

Multiplique $- \frac{3}{5} \cdot 0$ .

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Etapa 4.2.1.2.2.1

Multiplique $0$ por $- 1$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{3}{5} & 0 (\frac{3}{5}) & - \frac{3}{5} \cdot 0 \\ - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 1 & - \frac{3}{5} \cdot 0 \\ - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 1 \end{array}]$

Etapa 4.2.1.2.2.2

Multiplique $0$ por $\frac{3}{5}$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{3}{5} & 0 & - \frac{3}{5} \cdot 0 \\ - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 1 & - \frac{3}{5} \cdot 0 \\ - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 1 \end{array}]$

Etapa 4.2.1.2.3

Multiplique $- \frac{3}{5} \cdot 0$ .

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Etapa 4.2.1.2.3.1

Multiplique $0$ por $- 1$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{3}{5} & 0 & 0 (\frac{3}{5}) \\ - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 1 & - \frac{3}{5} \cdot 0 \\ - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 1 \end{array}]$

Etapa 4.2.1.2.3.2

Multiplique $0$ por $\frac{3}{5}$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{3}{5} & 0 & 0 \\ - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 1 & - \frac{3}{5} \cdot 0 \\ - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 1 \end{array}]$

Etapa 4.2.1.2.4

Multiplique $- \frac{3}{5} \cdot 0$ .

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Etapa 4.2.1.2.4.1

Multiplique $0$ por $- 1$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{3}{5} & 0 & 0 \\ 0 (\frac{3}{5}) & - \frac{3}{5} \cdot 1 & - \frac{3}{5} \cdot 0 \\ - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 1 \end{array}]$

Etapa 4.2.1.2.4.2

Multiplique $0$ por $\frac{3}{5}$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{3}{5} & 0 & 0 \\ 0 & - \frac{3}{5} \cdot 1 & - \frac{3}{5} \cdot 0 \\ - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 1 \end{array}]$

Etapa 4.2.1.2.5

Multiplique $- 1$ por $1$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{3}{5} & 0 & 0 \\ 0 & - \frac{3}{5} & - \frac{3}{5} \cdot 0 \\ - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 1 \end{array}]$

Etapa 4.2.1.2.6

Multiplique $- \frac{3}{5} \cdot 0$ .

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Etapa 4.2.1.2.6.1

Multiplique $0$ por $- 1$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{3}{5} & 0 & 0 \\ 0 & - \frac{3}{5} & 0 (\frac{3}{5}) \\ - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 1 \end{array}]$

Etapa 4.2.1.2.6.2

Multiplique $0$ por $\frac{3}{5}$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{3}{5} & 0 & 0 \\ 0 & - \frac{3}{5} & 0 \\ - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 1 \end{array}]$

Etapa 4.2.1.2.7

Multiplique $- \frac{3}{5} \cdot 0$ .

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Etapa 4.2.1.2.7.1

Multiplique $0$ por $- 1$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{3}{5} & 0 & 0 \\ 0 & - \frac{3}{5} & 0 \\ 0 (\frac{3}{5}) & - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 1 \end{array}]$

Etapa 4.2.1.2.7.2

Multiplique $0$ por $\frac{3}{5}$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{3}{5} & 0 & 0 \\ 0 & - \frac{3}{5} & 0 \\ 0 & - \frac{3}{5} \cdot 0 & - \frac{3}{5} \cdot 1 \end{array}]$

Etapa 4.2.1.2.8

Multiplique $- \frac{3}{5} \cdot 0$ .

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Etapa 4.2.1.2.8.1

Multiplique $0$ por $- 1$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{3}{5} & 0 & 0 \\ 0 & - \frac{3}{5} & 0 \\ 0 & 0 (\frac{3}{5}) & - \frac{3}{5} \cdot 1 \end{array}]$

Etapa 4.2.1.2.8.2

Multiplique $0$ por $\frac{3}{5}$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{3}{5} & 0 & 0 \\ 0 & - \frac{3}{5} & 0 \\ 0 & 0 & - \frac{3}{5} \cdot 1 \end{array}]$

Etapa 4.2.1.2.9

Multiplique $- 1$ por $1$ .

$[\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] + [\begin{array}{c} - \frac{3}{5} & 0 & 0 \\ 0 & - \frac{3}{5} & 0 \\ 0 & 0 & - \frac{3}{5} \end{array}]$

Etapa 4.2.2

Adicione os elementos correspondentes.

$[\begin{array}{c} 0.8 - \frac{3}{5} & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{3}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3

Simplify each element.

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Etapa 4.2.3.1

Para escrever $0.8$ como fração com um denominador comum, multiplique por $\frac{5}{5}$ .

$[\begin{array}{c} 0.8 \cdot \frac{5}{5} - \frac{3}{5} & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{3}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.2

Combine $0.8$ e $\frac{5}{5}$ .

$[\begin{array}{c} \frac{0.8 \cdot 5}{5} - \frac{3}{5} & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{3}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.3

Combine os numeradores em relação ao denominador comum.

$[\begin{array}{c} \frac{0.8 \cdot 5 - 3}{5} & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{3}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.4

Simplifique o numerador.

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Etapa 4.2.3.4.1

Multiplique $0.8$ por $5$ .

$[\begin{array}{c} \frac{4 - 3}{5} & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{3}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.4.2

Subtraia $3$ de $4$ .

$[\begin{array}{c} \frac{1}{5} & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{3}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.5

Cancele o fator comum de $1$ e $5$ .

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Etapa 4.2.3.5.1

Reescreva $1$ como $1 (1)$ .

$[\begin{array}{c} \frac{1 (1)}{5} & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{3}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.5.2

Cancele os fatores comuns.

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Etapa 4.2.3.5.2.1

Reescreva $5$ como $1 (5)$ .

$[\begin{array}{c} \frac{1 \cdot 1}{1 \cdot 5} & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{3}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.5.2.2

Cancele o fator comum.

$[\begin{array}{c} \frac{1 \cdot 1}{1 \cdot 5} & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{3}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.5.2.3

Reescreva a expressão.

$[\begin{array}{c} \frac{1}{5} & 0.2 + 0 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{3}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.6

Some $0.2$ e $0$ .

$[\begin{array}{c} \frac{1}{5} & 0.2 & 0.2 + 0 \\ 0 + 0 & 0.5 - \frac{3}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.7

Some $0.2$ e $0$ .

$[\begin{array}{c} \frac{1}{5} & 0.2 & 0.2 \\ 0 + 0 & 0.5 - \frac{3}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.8

Some $0$ e $0$ .

$[\begin{array}{c} \frac{1}{5} & 0.2 & 0.2 \\ 0 & 0.5 - \frac{3}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.9

Para escrever $0.5$ como fração com um denominador comum, multiplique por $\frac{5}{5}$ .

$[\begin{array}{c} \frac{1}{5} & 0.2 & 0.2 \\ 0 & 0.5 \cdot \frac{5}{5} - \frac{3}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.10

Combine $0.5$ e $\frac{5}{5}$ .

$[\begin{array}{c} \frac{1}{5} & 0.2 & 0.2 \\ 0 & \frac{0.5 \cdot 5}{5} - \frac{3}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.11

Combine os numeradores em relação ao denominador comum.

$[\begin{array}{c} \frac{1}{5} & 0.2 & 0.2 \\ 0 & \frac{0.5 \cdot 5 - 3}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.12

Simplifique o numerador.

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Etapa 4.2.3.12.1

Multiplique $0.5$ por $5$ .

$[\begin{array}{c} \frac{1}{5} & 0.2 & 0.2 \\ 0 & \frac{2.5 - 3}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.12.2

Subtraia $3$ de $2.5$ .

$[\begin{array}{c} \frac{1}{5} & 0.2 & 0.2 \\ 0 & \frac{- 0.5}{5} & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.13

Divida $- 0.5$ por $5$ .

$[\begin{array}{c} \frac{1}{5} & 0.2 & 0.2 \\ 0 & - 0.1 & 0.1 + 0 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.14

Some $0.1$ e $0$ .

$[\begin{array}{c} \frac{1}{5} & 0.2 & 0.2 \\ 0 & - 0.1 & 0.1 \\ 0.2 + 0 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.15

Some $0.2$ e $0$ .

$[\begin{array}{c} \frac{1}{5} & 0.2 & 0.2 \\ 0 & - 0.1 & 0.1 \\ 0.2 & 0.3 + 0 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.16

Some $0.3$ e $0$ .

$[\begin{array}{c} \frac{1}{5} & 0.2 & 0.2 \\ 0 & - 0.1 & 0.1 \\ 0.2 & 0.3 & 0.7 - \frac{3}{5} \end{array}]$

Etapa 4.2.3.17

Para escrever $0.7$ como fração com um denominador comum, multiplique por $\frac{5}{5}$ .

$[\begin{array}{c} \frac{1}{5} & 0.2 & 0.2 \\ 0 & - 0.1 & 0.1 \\ 0.2 & 0.3 & 0.7 \cdot \frac{5}{5} - \frac{3}{5} \end{array}]$

Etapa 4.2.3.18

Combine $0.7$ e $\frac{5}{5}$ .

$[\begin{array}{c} \frac{1}{5} & 0.2 & 0.2 \\ 0 & - 0.1 & 0.1 \\ 0.2 & 0.3 & \frac{0.7 \cdot 5}{5} - \frac{3}{5} \end{array}]$

Etapa 4.2.3.19

Combine os numeradores em relação ao denominador comum.

$[\begin{array}{c} \frac{1}{5} & 0.2 & 0.2 \\ 0 & - 0.1 & 0.1 \\ 0.2 & 0.3 & \frac{0.7 \cdot 5 - 3}{5} \end{array}]$

Etapa 4.2.3.20

Simplifique o numerador.

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Etapa 4.2.3.20.1

Multiplique $0.7$ por $5$ .

$[\begin{array}{c} \frac{1}{5} & 0.2 & 0.2 \\ 0 & - 0.1 & 0.1 \\ 0.2 & 0.3 & \frac{3.5 - 3}{5} \end{array}]$

Etapa 4.2.3.20.2

Subtraia $3$ de $3.5$ .

$[\begin{array}{c} \frac{1}{5} & 0.2 & 0.2 \\ 0 & - 0.1 & 0.1 \\ 0.2 & 0.3 & \frac{0.5}{5} \end{array}]$

Etapa 4.2.3.21

Divida $0.5$ por $5$ .

$[\begin{array}{c} \frac{1}{5} & 0.2 & 0.2 \\ 0 & - 0.1 & 0.1 \\ 0.2 & 0.3 & 0.1 \end{array}]$

Etapa 4.3

Find the null space when $λ = \frac{3}{5}$ .

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Etapa 4.3.1

Write as an augmented matrix for $A x = 0$ .

$[\begin{array}{c} \frac{1}{5} & 0.2 & 0.2 & 0 \\ 0 & - 0.1 & 0.1 & 0 \\ 0.2 & 0.3 & 0.1 & 0 \end{array}]$

Etapa 4.3.2

Encontre a forma escalonada reduzida por linhas.

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Etapa 4.3.2.1

Multiply each element of $R_{1}$ by $5$ to make the entry at $1, 1$ a $1$ .

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Etapa 4.3.2.1.1

Multiply each element of $R_{1}$ by $5$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} 5 (\frac{1}{5}) & 5 \cdot 0.2 & 5 \cdot 0.2 & 5 \cdot 0 \\ 0 & - 0.1 & 0.1 & 0 \\ 0.2 & 0.3 & 0.1 & 0 \end{array}]$

Etapa 4.3.2.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 \\ 0 & - 0.1 & 0.1 & 0 \\ 0.2 & 0.3 & 0.1 & 0 \end{array}]$

Etapa 4.3.2.2

Perform the row operation $R_{3} = R_{3} - 0.2 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Etapa 4.3.2.2.1

Perform the row operation $R_{3} = R_{3} - 0.2 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 \\ 0 & - 0.1 & 0.1 & 0 \\ 0.2 - 0.2 \cdot 1 & 0.3 - 0.2 \cdot 1 & 0.1 - 0.2 \cdot 1 & 0 - 0.2 \cdot 0 \end{array}]$

Etapa 4.3.2.2.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 \\ 0 & - 0.1 & 0.1 & 0 \\ 0 & 0.1 & - 0.1 & 0 \end{array}]$

Etapa 4.3.2.3

Multiply each element of $R_{2}$ by $\frac{1}{- 0.1}$ to make the entry at $2, 2$ a $1$ .

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Etapa 4.3.2.3.1

Multiply each element of $R_{2}$ by $\frac{1}{- 0.1}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 \\ \frac{0}{- 0.1} & \frac{- 0.1}{- 0.1} & \frac{0.1}{- 0.1} & \frac{0}{- 0.1} \\ 0 & 0.1 & - 0.1 & 0 \end{array}]$

Etapa 4.3.2.3.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 \\ 0 & 1 & - 1 & 0 \\ 0 & 0.1 & - 0.1 & 0 \end{array}]$

Etapa 4.3.2.4

Perform the row operation $R_{3} = R_{3} - 0.1 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Etapa 4.3.2.4.1

Perform the row operation $R_{3} = R_{3} - 0.1 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 \\ 0 & 1 & - 1 & 0 \\ 0 - 0.1 \cdot 0 & 0.1 - 0.1 \cdot 1 & - 0.1 - 0.1 \cdot - 1 & 0 - 0.1 \cdot 0 \end{array}]$

Etapa 4.3.2.4.2

Simplifique $R_{3}$ .

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

Etapa 4.3.2.5

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

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Etapa 4.3.2.5.1

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

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

Etapa 4.3.2.5.2

Simplifique $R_{1}$ .

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

Etapa 4.3.3

Use the result matrix to declare the final solution to the system of equations.

$x + 2 z = 0$

$y - z = 0$

$0 = 0$

Etapa 4.3.4

Write a solution vector by solving in terms of the free variables in each row.

$[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} - 2 z \\ z \\ z \end{array}]$

Etapa 4.3.5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x \\ y \\ z \end{array}] = z [\begin{array}{c} - 2 \\ 1 \\ 1 \end{array}]$

Etapa 4.3.6

Write as a solution set.

${z [\begin{array}{c} - 2 \\ 1 \\ 1 \end{array}] | z \in R}$

Etapa 4.3.7

The solution is the set of vectors created from the free variables of the system.

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

Etapa 5

Find the eigenvector using the eigenvalue $λ = 1$ .

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Etapa 5.1

Substitua os valores conhecidos na fórmula.

$N ([\begin{array}{c} 0.8 & 0.2 & 0.2 \\ 0 & 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 \end{array}] - [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Etapa 5.2

Simplifique.

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Etapa 5.2.1

Subtraia os elementos correspondentes.

$[\begin{array}{c} 0.8 - 1 & 0.2 - 0 & 0.2 - 0 \\ 0 - 0 & 0.5 - 1 & 0.1 - 0 \\ 0.2 - 0 & 0.3 - 0 & 0.7 - 1 \end{array}]$

Etapa 5.2.2

Simplify each element.

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Etapa 5.2.2.1

Subtraia $1$ de $0.8$ .

$[\begin{array}{c} - 0.2 & 0.2 - 0 & 0.2 - 0 \\ 0 - 0 & 0.5 - 1 & 0.1 - 0 \\ 0.2 - 0 & 0.3 - 0 & 0.7 - 1 \end{array}]$

Etapa 5.2.2.2

Subtraia $0$ de $0.2$ .

$[\begin{array}{c} - 0.2 & 0.2 & 0.2 - 0 \\ 0 - 0 & 0.5 - 1 & 0.1 - 0 \\ 0.2 - 0 & 0.3 - 0 & 0.7 - 1 \end{array}]$

Etapa 5.2.2.3

Subtraia $0$ de $0.2$ .

$[\begin{array}{c} - 0.2 & 0.2 & 0.2 \\ 0 - 0 & 0.5 - 1 & 0.1 - 0 \\ 0.2 - 0 & 0.3 - 0 & 0.7 - 1 \end{array}]$

Etapa 5.2.2.4

Subtraia $0$ de $0$ .

$[\begin{array}{c} - 0.2 & 0.2 & 0.2 \\ 0 & 0.5 - 1 & 0.1 - 0 \\ 0.2 - 0 & 0.3 - 0 & 0.7 - 1 \end{array}]$

Etapa 5.2.2.5

Subtraia $1$ de $0.5$ .

$[\begin{array}{c} - 0.2 & 0.2 & 0.2 \\ 0 & - 0.5 & 0.1 - 0 \\ 0.2 - 0 & 0.3 - 0 & 0.7 - 1 \end{array}]$

Etapa 5.2.2.6

Subtraia $0$ de $0.1$ .

$[\begin{array}{c} - 0.2 & 0.2 & 0.2 \\ 0 & - 0.5 & 0.1 \\ 0.2 - 0 & 0.3 - 0 & 0.7 - 1 \end{array}]$

Etapa 5.2.2.7

Subtraia $0$ de $0.2$ .

$[\begin{array}{c} - 0.2 & 0.2 & 0.2 \\ 0 & - 0.5 & 0.1 \\ 0.2 & 0.3 - 0 & 0.7 - 1 \end{array}]$

Etapa 5.2.2.8

Subtraia $0$ de $0.3$ .

$[\begin{array}{c} - 0.2 & 0.2 & 0.2 \\ 0 & - 0.5 & 0.1 \\ 0.2 & 0.3 & 0.7 - 1 \end{array}]$

Etapa 5.2.2.9

Subtraia $1$ de $0.7$ .

$[\begin{array}{c} - 0.2 & 0.2 & 0.2 \\ 0 & - 0.5 & 0.1 \\ 0.2 & 0.3 & - 0.3 \end{array}]$

Etapa 5.3

Find the null space when $λ = 1$ .

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Etapa 5.3.1

Write as an augmented matrix for $A x = 0$ .

$[\begin{array}{c} - 0.2 & 0.2 & 0.2 & 0 \\ 0 & - 0.5 & 0.1 & 0 \\ 0.2 & 0.3 & - 0.3 & 0 \end{array}]$

Etapa 5.3.2

Encontre a forma escalonada reduzida por linhas.

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Etapa 5.3.2.1

Multiply each element of $R_{1}$ by $\frac{1}{- 0.2}$ to make the entry at $1, 1$ a $1$ .

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Etapa 5.3.2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{- 0.2}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{- 0.2}{- 0.2} & \frac{0.2}{- 0.2} & \frac{0.2}{- 0.2} & \frac{0}{- 0.2} \\ 0 & - 0.5 & 0.1 & 0 \\ 0.2 & 0.3 & - 0.3 & 0 \end{array}]$

Etapa 5.3.2.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 0 \\ 0 & - 0.5 & 0.1 & 0 \\ 0.2 & 0.3 & - 0.3 & 0 \end{array}]$

Etapa 5.3.2.2

Perform the row operation $R_{3} = R_{3} - 0.2 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Etapa 5.3.2.2.1

Perform the row operation $R_{3} = R_{3} - 0.2 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 0 \\ 0 & - 0.5 & 0.1 & 0 \\ 0.2 - 0.2 \cdot 1 & 0.3 - 0.2 \cdot - 1 & - 0.3 - 0.2 \cdot - 1 & 0 - 0.2 \cdot 0 \end{array}]$

Etapa 5.3.2.2.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 0 \\ 0 & - 0.5 & 0.1 & 0 \\ 0 & 0.5 & - 0.1 & 0 \end{array}]$

Etapa 5.3.2.3

Multiply each element of $R_{2}$ by $\frac{1}{- 0.5}$ to make the entry at $2, 2$ a $1$ .

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Etapa 5.3.2.3.1

Multiply each element of $R_{2}$ by $\frac{1}{- 0.5}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 0 \\ \frac{0}{- 0.5} & \frac{- 0.5}{- 0.5} & \frac{0.1}{- 0.5} & \frac{0}{- 0.5} \\ 0 & 0.5 & - 0.1 & 0 \end{array}]$

Etapa 5.3.2.3.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 0 \\ 0 & 1 & - 0.2 & 0 \\ 0 & 0.5 & - 0.1 & 0 \end{array}]$

Etapa 5.3.2.4

Perform the row operation $R_{3} = R_{3} - 0.5 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Etapa 5.3.2.4.1

Perform the row operation $R_{3} = R_{3} - 0.5 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 0 \\ 0 & 1 & - 0.2 & 0 \\ 0 - 0.5 \cdot 0 & 0.5 - 0.5 \cdot 1 & - 0.1 - 0.5 \cdot - 0.2 & 0 - 0.5 \cdot 0 \end{array}]$

Etapa 5.3.2.4.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 0 \\ 0 & 1 & - 0.2 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 5.3.2.5

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

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Etapa 5.3.2.5.1

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

$[\begin{array}{c} 1 + 0 & - 1 + 1 \cdot 1 & - 1 - 0.2 & 0 + 0 \\ 0 & 1 & - 0.2 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 5.3.2.5.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - 1.2 & 0 \\ 0 & 1 & - 0.2 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 5.3.3

Use the result matrix to declare the final solution to the system of equations.

$x - 1.2 z = 0$

$y - 0.2 z = 0$

$0 = 0$

Etapa 5.3.4

Write a solution vector by solving in terms of the free variables in each row.

$[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 1.2 z \\ 0.2 z \\ z \end{array}]$

Etapa 5.3.5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x \\ y \\ z \end{array}] = z [\begin{array}{c} 1.2 \\ 0.2 \\ 1 \end{array}]$

Etapa 5.3.6

Write as a solution set.

${z [\begin{array}{c} 1.2 \\ 0.2 \\ 1 \end{array}] | z \in R}$

Etapa 5.3.7

The solution is the set of vectors created from the free variables of the system.

${[\begin{array}{c} 1.2 \\ 0.2 \\ 1 \end{array}]}$

Etapa 6

The eigenspace of $A$ is the list of the vector space for each eigenvalue.

${[\begin{array}{c} 0 \\ - 1 \\ 1 \end{array}], [\begin{array}{c} - 2 \\ 1 \\ 1 \end{array}], [\begin{array}{c} 1.2 \\ 0.2 \\ 1 \end{array}]}$