Álgebra linear Exemplos

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}]$

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 $[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}]$ por $A$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] - λ I_{3})$

Etapa 1.3.2

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

$p (λ) = determinante ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

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.

$p (λ) = determinante ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2

Simplifique cada elemento da matriz.

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Etapa 1.4.1.2.1

Multiplique $- 1$ por $1$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.2

Multiplique $- λ \cdot 0$ .

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Etapa 1.4.1.2.2.1

Multiplique $0$ por $- 1$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.2.2

Multiplique $0$ por $λ$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.3

Multiplique $- λ \cdot 0$ .

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Etapa 1.4.1.2.3.1

Multiplique $0$ por $- 1$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.3.2

Multiplique $0$ por $λ$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.4

Multiplique $- λ \cdot 0$ .

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Etapa 1.4.1.2.4.1

Multiplique $0$ por $- 1$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 λ & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.4.2

Multiplique $0$ por $λ$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.5

Multiplique $- 1$ por $1$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.6

Multiplique $- λ \cdot 0$ .

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Etapa 1.4.1.2.6.1

Multiplique $0$ por $- 1$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.6.2

Multiplique $0$ por $λ$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.7

Multiplique $- λ \cdot 0$ .

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Etapa 1.4.1.2.7.1

Multiplique $0$ por $- 1$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.7.2

Multiplique $0$ por $λ$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.8

Multiplique $- λ \cdot 0$ .

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Etapa 1.4.1.2.8.1

Multiplique $0$ por $- 1$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 λ & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.8.2

Multiplique $0$ por $λ$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.9

Multiplique $- 1$ por $1$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \end{array}])$

Etapa 1.4.2

Adicione os elementos correspondentes.

$p (λ) = determinante [\begin{array}{c} 1 - λ & - 1 + 0 & 0 + 0 \\ 2 + 0 & 2 - λ & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 - λ \end{array}]$

Etapa 1.4.3

Simplify each element.

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Etapa 1.4.3.1

Some $- 1$ e $0$ .

$p (λ) = determinante [\begin{array}{c} 1 - λ & - 1 & 0 + 0 \\ 2 + 0 & 2 - λ & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 - λ \end{array}]$

Etapa 1.4.3.2

Some $0$ e $0$ .

$p (λ) = determinante [\begin{array}{c} 1 - λ & - 1 & 0 \\ 2 + 0 & 2 - λ & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 - λ \end{array}]$

Etapa 1.4.3.3

Some $2$ e $0$ .

$p (λ) = determinante [\begin{array}{c} 1 - λ & - 1 & 0 \\ 2 & 2 - λ & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 - λ \end{array}]$

Etapa 1.4.3.4

Some $1$ e $0$ .

$p (λ) = determinante [\begin{array}{c} 1 - λ & - 1 & 0 \\ 2 & 2 - λ & 1 \\ 0 + 0 & 16 + 0 & - 1 - λ \end{array}]$

Etapa 1.4.3.5

Some $0$ e $0$ .

$p (λ) = determinante [\begin{array}{c} 1 - λ & - 1 & 0 \\ 2 & 2 - λ & 1 \\ 0 & 16 + 0 & - 1 - λ \end{array}]$

Etapa 1.4.3.6

Some $16$ e $0$ .

$p (λ) = determinante [\begin{array}{c} 1 - λ & - 1 & 0 \\ 2 & 2 - λ & 1 \\ 0 & 16 & - 1 - λ \end{array}]$

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 row $1$ 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.

$| \begin{array}{c} 2 - λ & 1 \\ 16 & - 1 - λ \end{array} |$

Etapa 1.5.1.4

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

$(1 - λ) | \begin{array}{c} 2 - λ & 1 \\ 16 & - 1 - λ \end{array} |$

Etapa 1.5.1.5

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

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

Etapa 1.5.1.6

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

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

Etapa 1.5.1.7

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

$| \begin{array}{c} 2 & 2 - λ \\ 0 & 16 \end{array} |$

Etapa 1.5.1.8

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

$0 | \begin{array}{c} 2 & 2 - λ \\ 0 & 16 \end{array} |$

Etapa 1.5.1.9

Add the terms together.

$p (λ) = (1 - λ) | \begin{array}{c} 2 - λ & 1 \\ 16 & - 1 - λ \end{array} | + 1 | \begin{array}{c} 2 & 1 \\ 0 & - 1 - λ \end{array} | + 0 | \begin{array}{c} 2 & 2 - λ \\ 0 & 16 \end{array} |$

Etapa 1.5.2

Multiplique $0$ por $| \begin{array}{c} 2 & 2 - λ \\ 0 & 16 \end{array} |$ .

$p (λ) = (1 - λ) | \begin{array}{c} 2 - λ & 1 \\ 16 & - 1 - λ \end{array} | + 1 | \begin{array}{c} 2 & 1 \\ 0 & - 1 - λ \end{array} | + 0$

Etapa 1.5.3

Avalie $| \begin{array}{c} 2 - λ & 1 \\ 16 & - 1 - λ \end{array} |$ .

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Etapa 1.5.3.1

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

$p (λ) = (1 - λ) ((2 - λ) (- 1 - λ) - 16 \cdot 1) + 1 | \begin{array}{c} 2 & 1 \\ 0 & - 1 - λ \end{array} | + 0$

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 $(2 - λ) (- 1 - λ)$ usando o método FOIL.

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Etapa 1.5.3.2.1.1.1

Aplique a propriedade distributiva.

$p (λ) = (1 - λ) (2 (- 1 - λ) - λ (- 1 - λ) - 16 \cdot 1) + 1 | \begin{array}{c} 2 & 1 \\ 0 & - 1 - λ \end{array} | + 0$

Etapa 1.5.3.2.1.1.2

Aplique a propriedade distributiva.

$p (λ) = (1 - λ) (2 \cdot - 1 + 2 (- λ) - λ (- 1 - λ) - 16 \cdot 1) + 1 | \begin{array}{c} 2 & 1 \\ 0 & - 1 - λ \end{array} | + 0$

Etapa 1.5.3.2.1.1.3

Aplique a propriedade distributiva.

$p (λ) = (1 - λ) (2 \cdot - 1 + 2 (- λ) - λ \cdot - 1 - λ (- λ) - 16 \cdot 1) + 1 | \begin{array}{c} 2 & 1 \\ 0 & - 1 - λ \end{array} | + 0$

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 $2$ por $- 1$ .

$p (λ) = (1 - λ) (- 2 + 2 (- λ) - λ \cdot - 1 - λ (- λ) - 16 \cdot 1) + 1 | \begin{array}{c} 2 & 1 \\ 0 & - 1 - λ \end{array} | + 0$

Etapa 1.5.3.2.1.2.1.2

Multiplique $- 1$ por $2$ .

$p (λ) = (1 - λ) (- 2 - 2 λ - λ \cdot - 1 - λ (- λ) - 16 \cdot 1) + 1 | \begin{array}{c} 2 & 1 \\ 0 & - 1 - λ \end{array} | + 0$

Etapa 1.5.3.2.1.2.1.3

Multiplique $- λ \cdot - 1$ .

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Etapa 1.5.3.2.1.2.1.3.1

Multiplique $- 1$ por $- 1$ .

$p (λ) = (1 - λ) (- 2 - 2 λ + 1 λ - λ (- λ) - 16 \cdot 1) + 1 | \begin{array}{c} 2 & 1 \\ 0 & - 1 - λ \end{array} | + 0$

Etapa 1.5.3.2.1.2.1.3.2

Multiplique $λ$ por $1$ .

$p (λ) = (1 - λ) (- 2 - 2 λ + λ - λ (- λ) - 16 \cdot 1) + 1 | \begin{array}{c} 2 & 1 \\ 0 & - 1 - λ \end{array} | + 0$

Etapa 1.5.3.2.1.2.1.4

Reescreva usando a propriedade comutativa da multiplicação.

$p (λ) = (1 - λ) (- 2 - 2 λ + λ - 1 \cdot - 1 λ \cdot λ - 16 \cdot 1) + 1 | \begin{array}{c} 2 & 1 \\ 0 & - 1 - λ \end{array} | + 0$

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 $λ$ .

$p (λ) = (1 - λ) (- 2 - 2 λ + λ - 1 \cdot - 1 (λ \cdot λ) - 16 \cdot 1) + 1 | \begin{array}{c} 2 & 1 \\ 0 & - 1 - λ \end{array} | + 0$

Etapa 1.5.3.2.1.2.1.5.2

Multiplique $λ$ por $λ$ .

$p (λ) = (1 - λ) (- 2 - 2 λ + λ - 1 \cdot - 1 λ^{2} - 16 \cdot 1) + 1 | \begin{array}{c} 2 & 1 \\ 0 & - 1 - λ \end{array} | + 0$

Etapa 1.5.3.2.1.2.1.6

Multiplique $- 1$ por $- 1$ .

$p (λ) = (1 - λ) (- 2 - 2 λ + λ + 1 λ^{2} - 16 \cdot 1) + 1 | \begin{array}{c} 2 & 1 \\ 0 & - 1 - λ \end{array} | + 0$

Etapa 1.5.3.2.1.2.1.7

Multiplique $λ^{2}$ por $1$ .

$p (λ) = (1 - λ) (- 2 - 2 λ + λ + λ^{2} - 16 \cdot 1) + 1 | \begin{array}{c} 2 & 1 \\ 0 & - 1 - λ \end{array} | + 0$

Etapa 1.5.3.2.1.2.2

Some $- 2 λ$ e $λ$ .

$p (λ) = (1 - λ) (- 2 - λ + λ^{2} - 16 \cdot 1) + 1 | \begin{array}{c} 2 & 1 \\ 0 & - 1 - λ \end{array} | + 0$

Etapa 1.5.3.2.1.3

Multiplique $- 16$ por $1$ .

$p (λ) = (1 - λ) (- 2 - λ + λ^{2} - 16) + 1 | \begin{array}{c} 2 & 1 \\ 0 & - 1 - λ \end{array} | + 0$

Etapa 1.5.3.2.2

Subtraia $16$ de $- 2$ .

$p (λ) = (1 - λ) (- λ + λ^{2} - 18) + 1 | \begin{array}{c} 2 & 1 \\ 0 & - 1 - λ \end{array} | + 0$

Etapa 1.5.3.2.3

Reordene $- λ$ e $λ^{2}$ .

$p (λ) = (1 - λ) (λ^{2} - λ - 18) + 1 | \begin{array}{c} 2 & 1 \\ 0 & - 1 - λ \end{array} | + 0$

Etapa 1.5.4

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

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Etapa 1.5.4.1

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

$p (λ) = (1 - λ) (λ^{2} - λ - 18) + 1 (2 (- 1 - λ) + 0 \cdot 1) + 0$

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

Aplique a propriedade distributiva.

$p (λ) = (1 - λ) (λ^{2} - λ - 18) + 1 (2 \cdot - 1 + 2 (- λ) + 0 \cdot 1) + 0$

Etapa 1.5.4.2.1.2

Multiplique $2$ por $- 1$ .

$p (λ) = (1 - λ) (λ^{2} - λ - 18) + 1 (- 2 + 2 (- λ) + 0 \cdot 1) + 0$

Etapa 1.5.4.2.1.3

Multiplique $- 1$ por $2$ .

$p (λ) = (1 - λ) (λ^{2} - λ - 18) + 1 (- 2 - 2 λ + 0 \cdot 1) + 0$

Etapa 1.5.4.2.1.4

Multiplique $0$ por $1$ .

$p (λ) = (1 - λ) (λ^{2} - λ - 18) + 1 (- 2 - 2 λ + 0) + 0$

Etapa 1.5.4.2.2

Some $- 2 - 2 λ$ e $0$ .

$p (λ) = (1 - λ) (λ^{2} - λ - 18) + 1 (- 2 - 2 λ) + 0$

Etapa 1.5.4.2.3

Reordene $- 2$ e $- 2 λ$ .

$p (λ) = (1 - λ) (λ^{2} - λ - 18) + 1 (- 2 λ - 2) + 0$

Etapa 1.5.5

Simplifique o determinante.

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Etapa 1.5.5.1

Some $(1 - λ) (λ^{2} - λ - 18) + 1 (- 2 λ - 2)$ e $0$ .

$p (λ) = (1 - λ) (λ^{2} - λ - 18) + 1 (- 2 λ - 2)$

Etapa 1.5.5.2

Simplifique cada termo.

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Etapa 1.5.5.2.1

Expanda $(1 - λ) (λ^{2} - λ - 18)$ multiplicando cada termo na primeira expressão por cada um dos termos na segunda expressão.

$p (λ) = 1 λ^{2} + 1 (- λ) + 1 \cdot - 18 - λ \cdot λ^{2} - λ (- λ) - λ \cdot - 18 + 1 (- 2 λ - 2)$

Etapa 1.5.5.2.2

Simplifique cada termo.

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Etapa 1.5.5.2.2.1

Multiplique $λ^{2}$ por $1$ .

$p (λ) = λ^{2} + 1 (- λ) + 1 \cdot - 18 - λ \cdot λ^{2} - λ (- λ) - λ \cdot - 18 + 1 (- 2 λ - 2)$

Etapa 1.5.5.2.2.2

Multiplique $- λ$ por $1$ .

$p (λ) = λ^{2} - λ + 1 \cdot - 18 - λ \cdot λ^{2} - λ (- λ) - λ \cdot - 18 + 1 (- 2 λ - 2)$

Etapa 1.5.5.2.2.3

Multiplique $- 18$ por $1$ .

$p (λ) = λ^{2} - λ - 18 - λ \cdot λ^{2} - λ (- λ) - λ \cdot - 18 + 1 (- 2 λ - 2)$

Etapa 1.5.5.2.2.4

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

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Etapa 1.5.5.2.2.4.1

Mova $λ^{2}$ .

$p (λ) = λ^{2} - λ - 18 - (λ^{2} λ) - λ (- λ) - λ \cdot - 18 + 1 (- 2 λ - 2)$

Etapa 1.5.5.2.2.4.2

Multiplique $λ^{2}$ por $λ$ .

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Etapa 1.5.5.2.2.4.2.1

Eleve $λ$ à potência de $1$ .

$p (λ) = λ^{2} - λ - 18 - (λ^{2} λ^{1}) - λ (- λ) - λ \cdot - 18 + 1 (- 2 λ - 2)$

Etapa 1.5.5.2.2.4.2.2

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

$p (λ) = λ^{2} - λ - 18 - λ^{2 + 1} - λ (- λ) - λ \cdot - 18 + 1 (- 2 λ - 2)$

Etapa 1.5.5.2.2.4.3

Some $2$ e $1$ .

$p (λ) = λ^{2} - λ - 18 - λ^{3} - λ (- λ) - λ \cdot - 18 + 1 (- 2 λ - 2)$

Etapa 1.5.5.2.2.5

Reescreva usando a propriedade comutativa da multiplicação.

$p (λ) = λ^{2} - λ - 18 - λ^{3} - 1 \cdot - 1 λ \cdot λ - λ \cdot - 18 + 1 (- 2 λ - 2)$

Etapa 1.5.5.2.2.6

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

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Etapa 1.5.5.2.2.6.1

Mova $λ$ .

$p (λ) = λ^{2} - λ - 18 - λ^{3} - 1 \cdot - 1 (λ \cdot λ) - λ \cdot - 18 + 1 (- 2 λ - 2)$

Etapa 1.5.5.2.2.6.2

Multiplique $λ$ por $λ$ .

$p (λ) = λ^{2} - λ - 18 - λ^{3} - 1 \cdot - 1 λ^{2} - λ \cdot - 18 + 1 (- 2 λ - 2)$

Etapa 1.5.5.2.2.7

Multiplique $- 1$ por $- 1$ .

$p (λ) = λ^{2} - λ - 18 - λ^{3} + 1 λ^{2} - λ \cdot - 18 + 1 (- 2 λ - 2)$

Etapa 1.5.5.2.2.8

Multiplique $λ^{2}$ por $1$ .

$p (λ) = λ^{2} - λ - 18 - λ^{3} + λ^{2} - λ \cdot - 18 + 1 (- 2 λ - 2)$

Etapa 1.5.5.2.2.9

Multiplique $- 18$ por $- 1$ .

$p (λ) = λ^{2} - λ - 18 - λ^{3} + λ^{2} + 18 λ + 1 (- 2 λ - 2)$

Etapa 1.5.5.2.3

Some $λ^{2}$ e $λ^{2}$ .

$p (λ) = 2 λ^{2} - λ - 18 - λ^{3} + 18 λ + 1 (- 2 λ - 2)$

Etapa 1.5.5.2.4

Some $- λ$ e $18 λ$ .

$p (λ) = 2 λ^{2} + 17 λ - 18 - λ^{3} + 1 (- 2 λ - 2)$

Etapa 1.5.5.2.5

Multiplique $- 2 λ - 2$ por $1$ .

$p (λ) = 2 λ^{2} + 17 λ - 18 - λ^{3} - 2 λ - 2$

Etapa 1.5.5.3

Subtraia $2 λ$ de $17 λ$ .

$p (λ) = 2 λ^{2} + 15 λ - 18 - λ^{3} - 2$

Etapa 1.5.5.4

Subtraia $2$ de $- 18$ .

$p (λ) = 2 λ^{2} + 15 λ - λ^{3} - 20$

Etapa 1.5.5.5

Mova $15 λ$ .

$p (λ) = 2 λ^{2} - λ^{3} + 15 λ - 20$

Etapa 1.5.5.6

Reordene $2 λ^{2}$ e $- λ^{3}$ .

$p (λ) = - λ^{3} + 2 λ^{2} + 15 λ - 20$

Etapa 1.6

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

$- λ^{3} + 2 λ^{2} + 15 λ - 20 = 0$

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.

$λ \approx - 3.63674676, 1.25510502, 4.38164173$

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 $λ = - 3.63674676$ .

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Etapa 3.1

Substitua os valores conhecidos na fórmula.

$N ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + 3.63674676 [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

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 $3.63674676$ por cada elemento da matriz.

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} 3.63674676 \cdot 1 & 3.63674676 \cdot 0 & 3.63674676 \cdot 0 \\ 3.63674676 \cdot 0 & 3.63674676 \cdot 1 & 3.63674676 \cdot 0 \\ 3.63674676 \cdot 0 & 3.63674676 \cdot 0 & 3.63674676 \cdot 1 \end{array}]$

Etapa 3.2.1.2

Simplifique cada elemento da matriz.

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Etapa 3.2.1.2.1

Multiplique $3.63674676$ por $1$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} 3.63674676 & 3.63674676 \cdot 0 & 3.63674676 \cdot 0 \\ 3.63674676 \cdot 0 & 3.63674676 \cdot 1 & 3.63674676 \cdot 0 \\ 3.63674676 \cdot 0 & 3.63674676 \cdot 0 & 3.63674676 \cdot 1 \end{array}]$

Etapa 3.2.1.2.2

Multiplique $3.63674676$ por $0$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} 3.63674676 & 0 & 3.63674676 \cdot 0 \\ 3.63674676 \cdot 0 & 3.63674676 \cdot 1 & 3.63674676 \cdot 0 \\ 3.63674676 \cdot 0 & 3.63674676 \cdot 0 & 3.63674676 \cdot 1 \end{array}]$

Etapa 3.2.1.2.3

Multiplique $3.63674676$ por $0$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} 3.63674676 & 0 & 0 \\ 3.63674676 \cdot 0 & 3.63674676 \cdot 1 & 3.63674676 \cdot 0 \\ 3.63674676 \cdot 0 & 3.63674676 \cdot 0 & 3.63674676 \cdot 1 \end{array}]$

Etapa 3.2.1.2.4

Multiplique $3.63674676$ por $0$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} 3.63674676 & 0 & 0 \\ 0 & 3.63674676 \cdot 1 & 3.63674676 \cdot 0 \\ 3.63674676 \cdot 0 & 3.63674676 \cdot 0 & 3.63674676 \cdot 1 \end{array}]$

Etapa 3.2.1.2.5

Multiplique $3.63674676$ por $1$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} 3.63674676 & 0 & 0 \\ 0 & 3.63674676 & 3.63674676 \cdot 0 \\ 3.63674676 \cdot 0 & 3.63674676 \cdot 0 & 3.63674676 \cdot 1 \end{array}]$

Etapa 3.2.1.2.6

Multiplique $3.63674676$ por $0$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} 3.63674676 & 0 & 0 \\ 0 & 3.63674676 & 0 \\ 3.63674676 \cdot 0 & 3.63674676 \cdot 0 & 3.63674676 \cdot 1 \end{array}]$

Etapa 3.2.1.2.7

Multiplique $3.63674676$ por $0$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} 3.63674676 & 0 & 0 \\ 0 & 3.63674676 & 0 \\ 0 & 3.63674676 \cdot 0 & 3.63674676 \cdot 1 \end{array}]$

Etapa 3.2.1.2.8

Multiplique $3.63674676$ por $0$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} 3.63674676 & 0 & 0 \\ 0 & 3.63674676 & 0 \\ 0 & 0 & 3.63674676 \cdot 1 \end{array}]$

Etapa 3.2.1.2.9

Multiplique $3.63674676$ por $1$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} 3.63674676 & 0 & 0 \\ 0 & 3.63674676 & 0 \\ 0 & 0 & 3.63674676 \end{array}]$

Etapa 3.2.2

Adicione os elementos correspondentes.

$[\begin{array}{c} 1 + 3.63674676 & - 1 + 0 & 0 + 0 \\ 2 + 0 & 2 + 3.63674676 & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 + 3.63674676 \end{array}]$

Etapa 3.2.3

Simplify each element.

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Etapa 3.2.3.1

Some $1$ e $3.63674676$ .

$[\begin{array}{c} 4.63674676 & - 1 + 0 & 0 + 0 \\ 2 + 0 & 2 + 3.63674676 & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 + 3.63674676 \end{array}]$

Etapa 3.2.3.2

Some $- 1$ e $0$ .

$[\begin{array}{c} 4.63674676 & - 1 & 0 + 0 \\ 2 + 0 & 2 + 3.63674676 & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 + 3.63674676 \end{array}]$

Etapa 3.2.3.3

Some $0$ e $0$ .

$[\begin{array}{c} 4.63674676 & - 1 & 0 \\ 2 + 0 & 2 + 3.63674676 & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 + 3.63674676 \end{array}]$

Etapa 3.2.3.4

Some $2$ e $0$ .

$[\begin{array}{c} 4.63674676 & - 1 & 0 \\ 2 & 2 + 3.63674676 & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 + 3.63674676 \end{array}]$

Etapa 3.2.3.5

Some $2$ e $3.63674676$ .

$[\begin{array}{c} 4.63674676 & - 1 & 0 \\ 2 & 5.63674676 & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 + 3.63674676 \end{array}]$

Etapa 3.2.3.6

Some $1$ e $0$ .

$[\begin{array}{c} 4.63674676 & - 1 & 0 \\ 2 & 5.63674676 & 1 \\ 0 + 0 & 16 + 0 & - 1 + 3.63674676 \end{array}]$

Etapa 3.2.3.7

Some $0$ e $0$ .

$[\begin{array}{c} 4.63674676 & - 1 & 0 \\ 2 & 5.63674676 & 1 \\ 0 & 16 + 0 & - 1 + 3.63674676 \end{array}]$

Etapa 3.2.3.8

Some $16$ e $0$ .

$[\begin{array}{c} 4.63674676 & - 1 & 0 \\ 2 & 5.63674676 & 1 \\ 0 & 16 & - 1 + 3.63674676 \end{array}]$

Etapa 3.2.3.9

Some $- 1$ e $3.63674676$ .

$[\begin{array}{c} 4.63674676 & - 1 & 0 \\ 2 & 5.63674676 & 1 \\ 0 & 16 & 2.63674676 \end{array}]$

Etapa 3.3

Find the null space when $λ = - 3.63674676$ .

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Etapa 3.3.1

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

$[\begin{array}{c} 4.63674676 & - 1 & 0 & 0 \\ 2 & 5.63674676 & 1 & 0 \\ 0 & 16 & 2.63674676 & 0 \end{array}]$

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 $\frac{1}{4.63674676}$ 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 $\frac{1}{4.63674676}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{4.63674676}{4.63674676} & \frac{- 1}{4.63674676} & \frac{0}{4.63674676} & \frac{0}{4.63674676} \\ 2 & 5.63674676 & 1 & 0 \\ 0 & 16 & 2.63674676 & 0 \end{array}]$

Etapa 3.3.2.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & - 0.21566845 & 0 & 0 \\ 2 & 5.63674676 & 1 & 0 \\ 0 & 16 & 2.63674676 & 0 \end{array}]$

Etapa 3.3.2.2

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

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Etapa 3.3.2.2.1

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

$[\begin{array}{c} 1 & - 0.21566845 & 0 & 0 \\ 2 - 2 \cdot 1 & 5.63674676 - 2 \cdot - 0.21566845 & 1 - 2 \cdot 0 & 0 - 2 \cdot 0 \\ 0 & 16 & 2.63674676 & 0 \end{array}]$

Etapa 3.3.2.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & - 0.21566845 & 0 & 0 \\ 0 & 6.06808367 & 1 & 0 \\ 0 & 16 & 2.63674676 & 0 \end{array}]$

Etapa 3.3.2.3

Multiply each element of $R_{2}$ by $\frac{1}{6.06808367}$ 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 $\frac{1}{6.06808367}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 0.21566845 & 0 & 0 \\ \frac{0}{6.06808367} & \frac{6.06808367}{6.06808367} & \frac{1}{6.06808367} & \frac{0}{6.06808367} \\ 0 & 16 & 2.63674676 & 0 \end{array}]$

Etapa 3.3.2.3.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & - 0.21566845 & 0 & 0 \\ 0 & 1 & 0.16479667 & 0 \\ 0 & 16 & 2.63674676 & 0 \end{array}]$

Etapa 3.3.2.4

Perform the row operation $R_{3} = R_{3} - 16 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Etapa 3.3.2.4.1

Perform the row operation $R_{3} = R_{3} - 16 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 0.21566845 & 0 & 0 \\ 0 & 1 & 0.16479667 & 0 \\ 0 - 16 \cdot 0 & 16 - 16 \cdot 1 & 2.63674676 - 16 \cdot 0.16479667 & 0 - 16 \cdot 0 \end{array}]$

Etapa 3.3.2.4.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & - 0.21566845 & 0 & 0 \\ 0 & 1 & 0.16479667 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 3.3.2.5

Multiply each element of $R_{3}$ by $\frac{1}{6.28618176 \cdot 10^{- 12}}$ to make the entry at $3, 3$ a $1$ .

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Etapa 3.3.2.5.1

Multiply each element of $R_{3}$ by $\frac{1}{6.28618176 \cdot 10^{- 12}}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - 0.21566845 & 0 & 0 \\ 0 & 1 & 0.16479667 & 0 \\ \frac{0}{6.28618176 \cdot 10^{- 12}} & \frac{0}{6.28618176 \cdot 10^{- 12}} & \frac{0}{6.28618176 \cdot 10^{- 12}} & \frac{0}{6.28618176 \cdot 10^{- 12}} \end{array}]$

Etapa 3.3.2.5.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & - 0.21566845 & 0 & 0 \\ 0 & 1 & 0.16479667 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Etapa 3.3.2.6

Perform the row operation $R_{2} = R_{2} - 0.16479667 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Etapa 3.3.2.6.1

Perform the row operation $R_{2} = R_{2} - 0.16479667 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - 0.21566845 & 0 & 0 \\ 0 - 0.16479667 \cdot 0 & 1 - 0.16479667 \cdot 0 & 0.16479667 - 0.16479667 \cdot 1 & 0 - 0.16479667 \cdot 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Etapa 3.3.2.6.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & - 0.21566845 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Etapa 3.3.2.7

Perform the row operation $R_{1} = R_{1} + 0.21566845 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Etapa 3.3.2.7.1

Perform the row operation $R_{1} = R_{1} + 0.21566845 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + 0.21566845 \cdot 0 & - 0.21566845 + 0.21566845 \cdot 1 & 0 + 0.21566845 \cdot 0 & 0 + 0.21566845 \cdot 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Etapa 3.3.2.7.2

Simplifique $R_{1}$ .

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

Etapa 3.3.3

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

$x = 0$

$y = 0$

$z = 0$

Etapa 3.3.4

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

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

Etapa 3.3.5

Write as a solution set.

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

Etapa 4

Find the eigenvector using the eigenvalue $λ = 1.25510502$ .

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Etapa 4.1

Substitua os valores conhecidos na fórmula.

$N ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] - 1.25510502 [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

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 $- 1.25510502$ por cada elemento da matriz.

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - 1.25510502 \cdot 1 & - 1.25510502 \cdot 0 & - 1.25510502 \cdot 0 \\ - 1.25510502 \cdot 0 & - 1.25510502 \cdot 1 & - 1.25510502 \cdot 0 \\ - 1.25510502 \cdot 0 & - 1.25510502 \cdot 0 & - 1.25510502 \cdot 1 \end{array}]$

Etapa 4.2.1.2

Simplifique cada elemento da matriz.

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Etapa 4.2.1.2.1

Multiplique $- 1.25510502$ por $1$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - 1.25510502 & - 1.25510502 \cdot 0 & - 1.25510502 \cdot 0 \\ - 1.25510502 \cdot 0 & - 1.25510502 \cdot 1 & - 1.25510502 \cdot 0 \\ - 1.25510502 \cdot 0 & - 1.25510502 \cdot 0 & - 1.25510502 \cdot 1 \end{array}]$

Etapa 4.2.1.2.2

Multiplique $- 1.25510502$ por $0$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - 1.25510502 & 0 & - 1.25510502 \cdot 0 \\ - 1.25510502 \cdot 0 & - 1.25510502 \cdot 1 & - 1.25510502 \cdot 0 \\ - 1.25510502 \cdot 0 & - 1.25510502 \cdot 0 & - 1.25510502 \cdot 1 \end{array}]$

Etapa 4.2.1.2.3

Multiplique $- 1.25510502$ por $0$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - 1.25510502 & 0 & 0 \\ - 1.25510502 \cdot 0 & - 1.25510502 \cdot 1 & - 1.25510502 \cdot 0 \\ - 1.25510502 \cdot 0 & - 1.25510502 \cdot 0 & - 1.25510502 \cdot 1 \end{array}]$

Etapa 4.2.1.2.4

Multiplique $- 1.25510502$ por $0$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - 1.25510502 & 0 & 0 \\ 0 & - 1.25510502 \cdot 1 & - 1.25510502 \cdot 0 \\ - 1.25510502 \cdot 0 & - 1.25510502 \cdot 0 & - 1.25510502 \cdot 1 \end{array}]$

Etapa 4.2.1.2.5

Multiplique $- 1.25510502$ por $1$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - 1.25510502 & 0 & 0 \\ 0 & - 1.25510502 & - 1.25510502 \cdot 0 \\ - 1.25510502 \cdot 0 & - 1.25510502 \cdot 0 & - 1.25510502 \cdot 1 \end{array}]$

Etapa 4.2.1.2.6

Multiplique $- 1.25510502$ por $0$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - 1.25510502 & 0 & 0 \\ 0 & - 1.25510502 & 0 \\ - 1.25510502 \cdot 0 & - 1.25510502 \cdot 0 & - 1.25510502 \cdot 1 \end{array}]$

Etapa 4.2.1.2.7

Multiplique $- 1.25510502$ por $0$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - 1.25510502 & 0 & 0 \\ 0 & - 1.25510502 & 0 \\ 0 & - 1.25510502 \cdot 0 & - 1.25510502 \cdot 1 \end{array}]$

Etapa 4.2.1.2.8

Multiplique $- 1.25510502$ por $0$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - 1.25510502 & 0 & 0 \\ 0 & - 1.25510502 & 0 \\ 0 & 0 & - 1.25510502 \cdot 1 \end{array}]$

Etapa 4.2.1.2.9

Multiplique $- 1.25510502$ por $1$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - 1.25510502 & 0 & 0 \\ 0 & - 1.25510502 & 0 \\ 0 & 0 & - 1.25510502 \end{array}]$

Etapa 4.2.2

Adicione os elementos correspondentes.

$[\begin{array}{c} 1 - 1.25510502 & - 1 + 0 & 0 + 0 \\ 2 + 0 & 2 - 1.25510502 & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 - 1.25510502 \end{array}]$

Etapa 4.2.3

Simplify each element.

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Etapa 4.2.3.1

Subtraia $1.25510502$ de $1$ .

$[\begin{array}{c} - 0.25510502 & - 1 + 0 & 0 + 0 \\ 2 + 0 & 2 - 1.25510502 & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 - 1.25510502 \end{array}]$

Etapa 4.2.3.2

Some $- 1$ e $0$ .

$[\begin{array}{c} - 0.25510502 & - 1 & 0 + 0 \\ 2 + 0 & 2 - 1.25510502 & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 - 1.25510502 \end{array}]$

Etapa 4.2.3.3

Some $0$ e $0$ .

$[\begin{array}{c} - 0.25510502 & - 1 & 0 \\ 2 + 0 & 2 - 1.25510502 & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 - 1.25510502 \end{array}]$

Etapa 4.2.3.4

Some $2$ e $0$ .

$[\begin{array}{c} - 0.25510502 & - 1 & 0 \\ 2 & 2 - 1.25510502 & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 - 1.25510502 \end{array}]$

Etapa 4.2.3.5

Subtraia $1.25510502$ de $2$ .

$[\begin{array}{c} - 0.25510502 & - 1 & 0 \\ 2 & 0.74489497 & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 - 1.25510502 \end{array}]$

Etapa 4.2.3.6

Some $1$ e $0$ .

$[\begin{array}{c} - 0.25510502 & - 1 & 0 \\ 2 & 0.74489497 & 1 \\ 0 + 0 & 16 + 0 & - 1 - 1.25510502 \end{array}]$

Etapa 4.2.3.7

Some $0$ e $0$ .

$[\begin{array}{c} - 0.25510502 & - 1 & 0 \\ 2 & 0.74489497 & 1 \\ 0 & 16 + 0 & - 1 - 1.25510502 \end{array}]$

Etapa 4.2.3.8

Some $16$ e $0$ .

$[\begin{array}{c} - 0.25510502 & - 1 & 0 \\ 2 & 0.74489497 & 1 \\ 0 & 16 & - 1 - 1.25510502 \end{array}]$

Etapa 4.2.3.9

Subtraia $1.25510502$ de $- 1$ .

$[\begin{array}{c} - 0.25510502 & - 1 & 0 \\ 2 & 0.74489497 & 1 \\ 0 & 16 & - 2.25510502 \end{array}]$

Etapa 4.3

Find the null space when $λ = 1.25510502$ .

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Etapa 4.3.1

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

$[\begin{array}{c} - 0.25510502 & - 1 & 0 & 0 \\ 2 & 0.74489497 & 1 & 0 \\ 0 & 16 & - 2.25510502 & 0 \end{array}]$

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 $\frac{1}{- 0.25510502}$ 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 $\frac{1}{- 0.25510502}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{- 0.25510502}{- 0.25510502} & \frac{- 1}{- 0.25510502} & \frac{0}{- 0.25510502} & \frac{0}{- 0.25510502} \\ 2 & 0.74489497 & 1 & 0 \\ 0 & 16 & - 2.25510502 & 0 \end{array}]$

Etapa 4.3.2.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 3.91995409 & 0 & 0 \\ 2 & 0.74489497 & 1 & 0 \\ 0 & 16 & - 2.25510502 & 0 \end{array}]$

Etapa 4.3.2.2

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

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Etapa 4.3.2.2.1

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

$[\begin{array}{c} 1 & 3.91995409 & 0 & 0 \\ 2 - 2 \cdot 1 & 0.74489497 - 2 \cdot 3.91995409 & 1 - 2 \cdot 0 & 0 - 2 \cdot 0 \\ 0 & 16 & - 2.25510502 & 0 \end{array}]$

Etapa 4.3.2.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 3.91995409 & 0 & 0 \\ 0 & - 7.09501322 & 1 & 0 \\ 0 & 16 & - 2.25510502 & 0 \end{array}]$

Etapa 4.3.2.3

Multiply each element of $R_{2}$ by $\frac{1}{- 7.09501322}$ 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 $\frac{1}{- 7.09501322}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 3.91995409 & 0 & 0 \\ \frac{0}{- 7.09501322} & \frac{- 7.09501322}{- 7.09501322} & \frac{1}{- 7.09501322} & \frac{0}{- 7.09501322} \\ 0 & 16 & - 2.25510502 & 0 \end{array}]$

Etapa 4.3.2.3.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 3.91995409 & 0 & 0 \\ 0 & 1 & - 0.14094406 & 0 \\ 0 & 16 & - 2.25510502 & 0 \end{array}]$

Etapa 4.3.2.4

Perform the row operation $R_{3} = R_{3} - 16 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Etapa 4.3.2.4.1

Perform the row operation $R_{3} = R_{3} - 16 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 3.91995409 & 0 & 0 \\ 0 & 1 & - 0.14094406 & 0 \\ 0 - 16 \cdot 0 & 16 - 16 \cdot 1 & - 2.25510502 - 16 \cdot - 0.14094406 & 0 - 16 \cdot 0 \end{array}]$

Etapa 4.3.2.4.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 3.91995409 & 0 & 0 \\ 0 & 1 & - 0.14094406 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 4.3.2.5

Multiply each element of $R_{3}$ by $\frac{1}{- 9.26597014 \cdot 10^{- 11}}$ to make the entry at $3, 3$ a $1$ .

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Etapa 4.3.2.5.1

Multiply each element of $R_{3}$ by $\frac{1}{- 9.26597014 \cdot 10^{- 11}}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 3.91995409 & 0 & 0 \\ 0 & 1 & - 0.14094406 & 0 \\ \frac{0}{- 9.26597014 \cdot 10^{- 11}} & \frac{0}{- 9.26597014 \cdot 10^{- 11}} & \frac{0}{- 9.26597014 \cdot 10^{- 11}} & \frac{0}{- 9.26597014 \cdot 10^{- 11}} \end{array}]$

Etapa 4.3.2.5.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 3.91995409 & 0 & 0 \\ 0 & 1 & - 0.14094406 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Etapa 4.3.2.6

Perform the row operation $R_{2} = R_{2} + 0.14094406 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Etapa 4.3.2.6.1

Perform the row operation $R_{2} = R_{2} + 0.14094406 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 3.91995409 & 0 & 0 \\ 0 + 0.14094406 \cdot 0 & 1 + 0.14094406 \cdot 0 & - 0.14094406 + 0.14094406 \cdot 1 & 0 + 0.14094406 \cdot 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Etapa 4.3.2.6.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 3.91995409 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Etapa 4.3.2.7

Perform the row operation $R_{1} = R_{1} - 3.91995409 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Etapa 4.3.2.7.1

Perform the row operation $R_{1} = R_{1} - 3.91995409 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 3.91995409 \cdot 0 & 3.91995409 - 3.91995409 \cdot 1 & 0 - 3.91995409 \cdot 0 & 0 - 3.91995409 \cdot 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Etapa 4.3.2.7.2

Simplifique $R_{1}$ .

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

Etapa 4.3.3

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

$x = 0$

$y = 0$

$z = 0$

Etapa 4.3.4

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

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

Etapa 4.3.5

Write as a solution set.

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

Etapa 5

Find the eigenvector using the eigenvalue $λ = 4.38164173$ .

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Etapa 5.1

Substitua os valores conhecidos na fórmula.

$N ([\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] - 4.38164173 [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Etapa 5.2

Simplifique.

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Etapa 5.2.1

Simplifique cada termo.

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Etapa 5.2.1.1

Multiplique $- 4.38164173$ por cada elemento da matriz.

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - 4.38164173 \cdot 1 & - 4.38164173 \cdot 0 & - 4.38164173 \cdot 0 \\ - 4.38164173 \cdot 0 & - 4.38164173 \cdot 1 & - 4.38164173 \cdot 0 \\ - 4.38164173 \cdot 0 & - 4.38164173 \cdot 0 & - 4.38164173 \cdot 1 \end{array}]$

Etapa 5.2.1.2

Simplifique cada elemento da matriz.

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Etapa 5.2.1.2.1

Multiplique $- 4.38164173$ por $1$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - 4.38164173 & - 4.38164173 \cdot 0 & - 4.38164173 \cdot 0 \\ - 4.38164173 \cdot 0 & - 4.38164173 \cdot 1 & - 4.38164173 \cdot 0 \\ - 4.38164173 \cdot 0 & - 4.38164173 \cdot 0 & - 4.38164173 \cdot 1 \end{array}]$

Etapa 5.2.1.2.2

Multiplique $- 4.38164173$ por $0$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - 4.38164173 & 0 & - 4.38164173 \cdot 0 \\ - 4.38164173 \cdot 0 & - 4.38164173 \cdot 1 & - 4.38164173 \cdot 0 \\ - 4.38164173 \cdot 0 & - 4.38164173 \cdot 0 & - 4.38164173 \cdot 1 \end{array}]$

Etapa 5.2.1.2.3

Multiplique $- 4.38164173$ por $0$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - 4.38164173 & 0 & 0 \\ - 4.38164173 \cdot 0 & - 4.38164173 \cdot 1 & - 4.38164173 \cdot 0 \\ - 4.38164173 \cdot 0 & - 4.38164173 \cdot 0 & - 4.38164173 \cdot 1 \end{array}]$

Etapa 5.2.1.2.4

Multiplique $- 4.38164173$ por $0$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - 4.38164173 & 0 & 0 \\ 0 & - 4.38164173 \cdot 1 & - 4.38164173 \cdot 0 \\ - 4.38164173 \cdot 0 & - 4.38164173 \cdot 0 & - 4.38164173 \cdot 1 \end{array}]$

Etapa 5.2.1.2.5

Multiplique $- 4.38164173$ por $1$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - 4.38164173 & 0 & 0 \\ 0 & - 4.38164173 & - 4.38164173 \cdot 0 \\ - 4.38164173 \cdot 0 & - 4.38164173 \cdot 0 & - 4.38164173 \cdot 1 \end{array}]$

Etapa 5.2.1.2.6

Multiplique $- 4.38164173$ por $0$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - 4.38164173 & 0 & 0 \\ 0 & - 4.38164173 & 0 \\ - 4.38164173 \cdot 0 & - 4.38164173 \cdot 0 & - 4.38164173 \cdot 1 \end{array}]$

Etapa 5.2.1.2.7

Multiplique $- 4.38164173$ por $0$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - 4.38164173 & 0 & 0 \\ 0 & - 4.38164173 & 0 \\ 0 & - 4.38164173 \cdot 0 & - 4.38164173 \cdot 1 \end{array}]$

Etapa 5.2.1.2.8

Multiplique $- 4.38164173$ por $0$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - 4.38164173 & 0 & 0 \\ 0 & - 4.38164173 & 0 \\ 0 & 0 & - 4.38164173 \cdot 1 \end{array}]$

Etapa 5.2.1.2.9

Multiplique $- 4.38164173$ por $1$ .

$[\begin{array}{c} 1 & - 1 & 0 \\ 2 & 2 & 1 \\ 0 & 16 & - 1 \end{array}] + [\begin{array}{c} - 4.38164173 & 0 & 0 \\ 0 & - 4.38164173 & 0 \\ 0 & 0 & - 4.38164173 \end{array}]$

Etapa 5.2.2

Adicione os elementos correspondentes.

$[\begin{array}{c} 1 - 4.38164173 & - 1 + 0 & 0 + 0 \\ 2 + 0 & 2 - 4.38164173 & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 - 4.38164173 \end{array}]$

Etapa 5.2.3

Simplify each element.

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Etapa 5.2.3.1

Subtraia $4.38164173$ de $1$ .

$[\begin{array}{c} - 3.38164173 & - 1 + 0 & 0 + 0 \\ 2 + 0 & 2 - 4.38164173 & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 - 4.38164173 \end{array}]$

Etapa 5.2.3.2

Some $- 1$ e $0$ .

$[\begin{array}{c} - 3.38164173 & - 1 & 0 + 0 \\ 2 + 0 & 2 - 4.38164173 & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 - 4.38164173 \end{array}]$

Etapa 5.2.3.3

Some $0$ e $0$ .

$[\begin{array}{c} - 3.38164173 & - 1 & 0 \\ 2 + 0 & 2 - 4.38164173 & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 - 4.38164173 \end{array}]$

Etapa 5.2.3.4

Some $2$ e $0$ .

$[\begin{array}{c} - 3.38164173 & - 1 & 0 \\ 2 & 2 - 4.38164173 & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 - 4.38164173 \end{array}]$

Etapa 5.2.3.5

Subtraia $4.38164173$ de $2$ .

$[\begin{array}{c} - 3.38164173 & - 1 & 0 \\ 2 & - 2.38164173 & 1 + 0 \\ 0 + 0 & 16 + 0 & - 1 - 4.38164173 \end{array}]$

Etapa 5.2.3.6

Some $1$ e $0$ .

$[\begin{array}{c} - 3.38164173 & - 1 & 0 \\ 2 & - 2.38164173 & 1 \\ 0 + 0 & 16 + 0 & - 1 - 4.38164173 \end{array}]$

Etapa 5.2.3.7

Some $0$ e $0$ .

$[\begin{array}{c} - 3.38164173 & - 1 & 0 \\ 2 & - 2.38164173 & 1 \\ 0 & 16 + 0 & - 1 - 4.38164173 \end{array}]$

Etapa 5.2.3.8

Some $16$ e $0$ .

$[\begin{array}{c} - 3.38164173 & - 1 & 0 \\ 2 & - 2.38164173 & 1 \\ 0 & 16 & - 1 - 4.38164173 \end{array}]$

Etapa 5.2.3.9

Subtraia $4.38164173$ de $- 1$ .

$[\begin{array}{c} - 3.38164173 & - 1 & 0 \\ 2 & - 2.38164173 & 1 \\ 0 & 16 & - 5.38164173 \end{array}]$

Etapa 5.3

Find the null space when $λ = 4.38164173$ .

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Etapa 5.3.1

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

$[\begin{array}{c} - 3.38164173 & - 1 & 0 & 0 \\ 2 & - 2.38164173 & 1 & 0 \\ 0 & 16 & - 5.38164173 & 0 \end{array}]$

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 $\frac{1}{- 3.38164173}$ 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 $\frac{1}{- 3.38164173}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{- 3.38164173}{- 3.38164173} & \frac{- 1}{- 3.38164173} & \frac{0}{- 3.38164173} & \frac{0}{- 3.38164173} \\ 2 & - 2.38164173 & 1 & 0 \\ 0 & 16 & - 5.38164173 & 0 \end{array}]$

Etapa 5.3.2.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 0.29571435 & 0 & 0 \\ 2 & - 2.38164173 & 1 & 0 \\ 0 & 16 & - 5.38164173 & 0 \end{array}]$

Etapa 5.3.2.2

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

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Etapa 5.3.2.2.1

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

$[\begin{array}{c} 1 & 0.29571435 & 0 & 0 \\ 2 - 2 \cdot 1 & - 2.38164173 - 2 \cdot 0.29571435 & 1 - 2 \cdot 0 & 0 - 2 \cdot 0 \\ 0 & 16 & - 5.38164173 & 0 \end{array}]$

Etapa 5.3.2.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 0.29571435 & 0 & 0 \\ 0 & - 2.97307044 & 1 & 0 \\ 0 & 16 & - 5.38164173 & 0 \end{array}]$

Etapa 5.3.2.3

Multiply each element of $R_{2}$ by $\frac{1}{- 2.97307044}$ 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 $\frac{1}{- 2.97307044}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 0.29571435 & 0 & 0 \\ \frac{0}{- 2.97307044} & \frac{- 2.97307044}{- 2.97307044} & \frac{1}{- 2.97307044} & \frac{0}{- 2.97307044} \\ 0 & 16 & - 5.38164173 & 0 \end{array}]$

Etapa 5.3.2.3.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 0.29571435 & 0 & 0 \\ 0 & 1 & - 0.3363526 & 0 \\ 0 & 16 & - 5.38164173 & 0 \end{array}]$

Etapa 5.3.2.4

Perform the row operation $R_{3} = R_{3} - 16 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Etapa 5.3.2.4.1

Perform the row operation $R_{3} = R_{3} - 16 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 0.29571435 & 0 & 0 \\ 0 & 1 & - 0.3363526 & 0 \\ 0 - 16 \cdot 0 & 16 - 16 \cdot 1 & - 5.38164173 - 16 \cdot - 0.3363526 & 0 - 16 \cdot 0 \end{array}]$

Etapa 5.3.2.4.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 0.29571435 & 0 & 0 \\ 0 & 1 & - 0.3363526 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 5.3.2.5

Multiply each element of $R_{3}$ by $\frac{1}{- 4.18780581 \cdot 10^{- 11}}$ to make the entry at $3, 3$ a $1$ .

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Etapa 5.3.2.5.1

Multiply each element of $R_{3}$ by $\frac{1}{- 4.18780581 \cdot 10^{- 11}}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 0.29571435 & 0 & 0 \\ 0 & 1 & - 0.3363526 & 0 \\ \frac{0}{- 4.18780581 \cdot 10^{- 11}} & \frac{0}{- 4.18780581 \cdot 10^{- 11}} & \frac{0}{- 4.18780581 \cdot 10^{- 11}} & \frac{0}{- 4.18780581 \cdot 10^{- 11}} \end{array}]$

Etapa 5.3.2.5.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 0.29571435 & 0 & 0 \\ 0 & 1 & - 0.3363526 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Etapa 5.3.2.6

Perform the row operation $R_{2} = R_{2} + 0.3363526 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Etapa 5.3.2.6.1

Perform the row operation $R_{2} = R_{2} + 0.3363526 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 0.29571435 & 0 & 0 \\ 0 + 0.3363526 \cdot 0 & 1 + 0.3363526 \cdot 0 & - 0.3363526 + 0.3363526 \cdot 1 & 0 + 0.3363526 \cdot 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Etapa 5.3.2.6.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 0.29571435 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Etapa 5.3.2.7

Perform the row operation $R_{1} = R_{1} - 0.29571435 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Etapa 5.3.2.7.1

Perform the row operation $R_{1} = R_{1} - 0.29571435 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 0.29571435 \cdot 0 & 0.29571435 - 0.29571435 \cdot 1 & 0 - 0.29571435 \cdot 0 & 0 - 0.29571435 \cdot 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Etapa 5.3.2.7.2

Simplifique $R_{1}$ .

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

Etapa 5.3.3

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

$x = 0$

$y = 0$

$z = 0$

Etapa 5.3.4

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

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

Etapa 5.3.5

Write as a solution set.

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

Etapa 6

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

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