Álgebra linear Exemplos

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

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 $[\begin{array}{c} - 3 & 0 & - 1 \\ 2 & 0 & 0 \\ 2 & 0 & 0 \end{array}]$ por $A$ .

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

Etapa 1.3.2

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

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

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.

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

Etapa 1.4.1.2

Simplifique cada elemento da matriz.

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Etapa 1.4.1.2.1

Multiplique $- 1$ por $1$ .

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

Etapa 1.4.1.2.2

Multiplique $- λ \cdot 0$ .

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Etapa 1.4.1.2.2.1

Multiplique $0$ por $- 1$ .

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

Etapa 1.4.1.2.2.2

Multiplique $0$ por $λ$ .

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

Etapa 1.4.1.2.3

Multiplique $- λ \cdot 0$ .

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Etapa 1.4.1.2.3.1

Multiplique $0$ por $- 1$ .

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

Etapa 1.4.1.2.3.2

Multiplique $0$ por $λ$ .

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

Etapa 1.4.1.2.4

Multiplique $- λ \cdot 0$ .

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Etapa 1.4.1.2.4.1

Multiplique $0$ por $- 1$ .

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

Etapa 1.4.1.2.4.2

Multiplique $0$ por $λ$ .

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

Etapa 1.4.1.2.5

Multiplique $- 1$ por $1$ .

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

Etapa 1.4.1.2.6

Multiplique $- λ \cdot 0$ .

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Etapa 1.4.1.2.6.1

Multiplique $0$ por $- 1$ .

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

Etapa 1.4.1.2.6.2

Multiplique $0$ por $λ$ .

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

Etapa 1.4.1.2.7

Multiplique $- λ \cdot 0$ .

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Etapa 1.4.1.2.7.1

Multiplique $0$ por $- 1$ .

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

Etapa 1.4.1.2.7.2

Multiplique $0$ por $λ$ .

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

Etapa 1.4.1.2.8

Multiplique $- λ \cdot 0$ .

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Etapa 1.4.1.2.8.1

Multiplique $0$ por $- 1$ .

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

Etapa 1.4.1.2.8.2

Multiplique $0$ por $λ$ .

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

Etapa 1.4.1.2.9

Multiplique $- 1$ por $1$ .

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

Etapa 1.4.2

Adicione os elementos correspondentes.

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

Etapa 1.4.3

Simplify each element.

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Etapa 1.4.3.1

Some $0$ e $0$ .

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

Etapa 1.4.3.2

Some $- 1$ e $0$ .

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

Etapa 1.4.3.3

Some $2$ e $0$ .

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

Etapa 1.4.3.4

Subtraia $λ$ de $0$ .

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

Etapa 1.4.3.5

Some $0$ e $0$ .

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

Etapa 1.4.3.6

Some $2$ e $0$ .

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

Etapa 1.4.3.7

Some $0$ e $0$ .

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

Etapa 1.4.3.8

Subtraia $λ$ de $0$ .

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

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

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

Etapa 1.5.1.4

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

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

Etapa 1.5.1.5

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

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

Etapa 1.5.1.6

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

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

Etapa 1.5.1.7

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

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

Etapa 1.5.1.8

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

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

Etapa 1.5.1.9

Add the terms together.

$p (λ) = 0 | \begin{array}{c} 2 & 0 \\ 2 & - λ \end{array} | - λ | \begin{array}{c} - 3 - λ & - 1 \\ 2 & - λ \end{array} | + 0 | \begin{array}{c} - 3 - λ & - 1 \\ 2 & 0 \end{array} |$

Etapa 1.5.2

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

$p (λ) = 0 - λ | \begin{array}{c} - 3 - λ & - 1 \\ 2 & - λ \end{array} | + 0 | \begin{array}{c} - 3 - λ & - 1 \\ 2 & 0 \end{array} |$

Etapa 1.5.3

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

$p (λ) = 0 - λ | \begin{array}{c} - 3 - λ & - 1 \\ 2 & - λ \end{array} | + 0$

Etapa 1.5.4

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

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Etapa 1.5.4.1

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

$p (λ) = 0 - λ ((- 3 - λ) (- λ) - 2 \cdot - 1) + 0$

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.

$p (λ) = 0 - λ (- 3 (- λ) - λ (- λ) - 2 \cdot - 1) + 0$

Etapa 1.5.4.2.1.2

Multiplique $- 1$ por $- 3$ .

$p (λ) = 0 - λ (3 λ - λ (- λ) - 2 \cdot - 1) + 0$

Etapa 1.5.4.2.1.3

Reescreva usando a propriedade comutativa da multiplicação.

$p (λ) = 0 - λ (3 λ - 1 \cdot - 1 λ \cdot λ - 2 \cdot - 1) + 0$

Etapa 1.5.4.2.1.4

Simplifique cada termo.

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Etapa 1.5.4.2.1.4.1

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

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Etapa 1.5.4.2.1.4.1.1

Mova $λ$ .

$p (λ) = 0 - λ (3 λ - 1 \cdot - 1 (λ \cdot λ) - 2 \cdot - 1) + 0$

Etapa 1.5.4.2.1.4.1.2

Multiplique $λ$ por $λ$ .

$p (λ) = 0 - λ (3 λ - 1 \cdot - 1 λ^{2} - 2 \cdot - 1) + 0$

Etapa 1.5.4.2.1.4.2

Multiplique $- 1$ por $- 1$ .

$p (λ) = 0 - λ (3 λ + 1 λ^{2} - 2 \cdot - 1) + 0$

Etapa 1.5.4.2.1.4.3

Multiplique $λ^{2}$ por $1$ .

$p (λ) = 0 - λ (3 λ + λ^{2} - 2 \cdot - 1) + 0$

Etapa 1.5.4.2.1.5

Multiplique $- 2$ por $- 1$ .

$p (λ) = 0 - λ (3 λ + λ^{2} + 2) + 0$

Etapa 1.5.4.2.2

Reordene $3 λ$ e $λ^{2}$ .

$p (λ) = 0 - λ (λ^{2} + 3 λ + 2) + 0$

Etapa 1.5.5

Simplifique o determinante.

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Etapa 1.5.5.1

Combine os termos opostos em $0 - λ (λ^{2} + 3 λ + 2) + 0$ .

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Etapa 1.5.5.1.1

Subtraia $λ (λ^{2} + 3 λ + 2)$ de $0$ .

$p (λ) = - λ (λ^{2} + 3 λ + 2) + 0$

Etapa 1.5.5.1.2

Some $- λ (λ^{2} + 3 λ + 2)$ e $0$ .

$p (λ) = - λ (λ^{2} + 3 λ + 2)$

Etapa 1.5.5.2

Aplique a propriedade distributiva.

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

Etapa 1.5.5.3

Simplifique.

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Etapa 1.5.5.3.1

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

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Etapa 1.5.5.3.1.1

Mova $λ^{2}$ .

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

Etapa 1.5.5.3.1.2

Multiplique $λ^{2}$ por $λ$ .

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Etapa 1.5.5.3.1.2.1

Eleve $λ$ à potência de $1$ .

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

Etapa 1.5.5.3.1.2.2

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

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

Etapa 1.5.5.3.1.3

Some $2$ e $1$ .

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

Etapa 1.5.5.3.2

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 1.5.5.3.3

Multiplique $2$ por $- 1$ .

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

Etapa 1.5.5.4

Simplifique cada termo.

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Etapa 1.5.5.4.1

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

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Etapa 1.5.5.4.1.1

Mova $λ$ .

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

Etapa 1.5.5.4.1.2

Multiplique $λ$ por $λ$ .

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

Etapa 1.5.5.4.2

Multiplique $- 1$ por $3$ .

$p (λ) = - λ^{3} - 3 λ^{2} - 2 λ$

Etapa 1.6

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

$- λ^{3} - 3 λ^{2} - 2 λ = 0$

Etapa 1.7

Resolva $λ$ .

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Etapa 1.7.1

Fatore o lado esquerdo da equação.

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Etapa 1.7.1.1

Fatore $- λ$ de $- λ^{3} - 3 λ^{2} - 2 λ$ .

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Etapa 1.7.1.1.1

Fatore $- λ$ de $- λ^{3}$ .

$- λ \cdot λ^{2} - 3 λ^{2} - 2 λ = 0$

Etapa 1.7.1.1.2

Fatore $- λ$ de $- 3 λ^{2}$ .

$- λ \cdot λ^{2} - λ (3 λ) - 2 λ = 0$

Etapa 1.7.1.1.3

Fatore $- λ$ de $- 2 λ$ .

$- λ \cdot λ^{2} - λ (3 λ) - λ \cdot 2 = 0$

Etapa 1.7.1.1.4

Fatore $- λ$ de $- λ (λ^{2}) - λ (3 λ)$ .

$- λ (λ^{2} + 3 λ) - λ \cdot 2 = 0$

Etapa 1.7.1.1.5

Fatore $- λ$ de $- λ (λ^{2} + 3 λ) - λ (2)$ .

$- λ (λ^{2} + 3 λ + 2) = 0$

Etapa 1.7.1.2

Fatore.

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Etapa 1.7.1.2.1

Fatore $λ^{2} + 3 λ + 2$ usando o método AC.

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Etapa 1.7.1.2.1.1

Considere a forma $x^{2} + b x + c$ . Encontre um par de números inteiros cujo produto é $c$ e cuja soma é $b$ . Neste caso, cujo produto é $2$ e cuja soma é $3$ .

$1, 2$

Etapa 1.7.1.2.1.2

Escreva a forma fatorada usando estes números inteiros.

$- λ ((λ + 1) (λ + 2)) = 0$

Etapa 1.7.1.2.2

Remova os parênteses desnecessários.

$- λ (λ + 1) (λ + 2) = 0$

Etapa 1.7.2

Se qualquer fator individual no lado esquerdo da equação for igual a $0$ , toda a expressão será igual a $0$ .

$λ = 0$

$λ + 1 = 0$

$λ + 2 = 0$

Etapa 1.7.3

Defina $λ$ como igual a $0$ .

$λ = 0$

Etapa 1.7.4

Defina $λ + 1$ como igual a $0$ e resolva para $λ$ .

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Etapa 1.7.4.1

Defina $λ + 1$ como igual a $0$ .

$λ + 1 = 0$

Etapa 1.7.4.2

Subtraia $1$ dos dois lados da equação.

$λ = - 1$

Etapa 1.7.5

Defina $λ + 2$ como igual a $0$ e resolva para $λ$ .

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Etapa 1.7.5.1

Defina $λ + 2$ como igual a $0$ .

$λ + 2 = 0$

Etapa 1.7.5.2

Subtraia $2$ dos dois lados da equação.

$λ = - 2$

Etapa 1.7.6

A solução final são todos os valores que tornam $- λ (λ + 1) (λ + 2) = 0$ verdadeiro.

$λ = 0, - 1, - 2$

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 $λ = 0$ .

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Etapa 3.1

Substitua os valores conhecidos na fórmula.

$N ([\begin{array}{c} - 3 & 0 & - 1 \\ 2 & 0 & 0 \\ 2 & 0 & 0 \end{array}] + 0 [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

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 $0$ por cada elemento da matriz.

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

Etapa 3.2.1.2

Simplifique cada elemento da matriz.

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Etapa 3.2.1.2.1

Multiplique $0$ por $1$ .

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

Etapa 3.2.1.2.2

Multiplique $0$ por $0$ .

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

Etapa 3.2.1.2.3

Multiplique $0$ por $0$ .

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

Etapa 3.2.1.2.4

Multiplique $0$ por $0$ .

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

Etapa 3.2.1.2.5

Multiplique $0$ por $1$ .

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

Etapa 3.2.1.2.6

Multiplique $0$ por $0$ .

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

Etapa 3.2.1.2.7

Multiplique $0$ por $0$ .

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

Etapa 3.2.1.2.8

Multiplique $0$ por $0$ .

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

Etapa 3.2.1.2.9

Multiplique $0$ por $1$ .

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

Etapa 3.2.2

Adding any matrix to a null matrix is the matrix itself.

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Etapa 3.2.2.1

Adicione os elementos correspondentes.

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

Etapa 3.2.2.2

Simplify each element.

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Etapa 3.2.2.2.1

Some $- 3$ e $0$ .

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

Etapa 3.2.2.2.2

Some $0$ e $0$ .

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

Etapa 3.2.2.2.3

Some $- 1$ e $0$ .

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

Etapa 3.2.2.2.4

Some $2$ e $0$ .

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

Etapa 3.2.2.2.5

Some $0$ e $0$ .

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

Etapa 3.2.2.2.6

Some $0$ e $0$ .

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

Etapa 3.2.2.2.7

Some $2$ e $0$ .

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

Etapa 3.2.2.2.8

Some $0$ e $0$ .

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

Etapa 3.2.2.2.9

Some $0$ e $0$ .

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

Etapa 3.3

Find the null space when $λ = 0$ .

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Etapa 3.3.1

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

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

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 $- \frac{1}{3}$ 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 $- \frac{1}{3}$ to make the entry at $1, 1$ a $1$ .

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

Etapa 3.3.2.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{3} & 0 \\ 2 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{3} & 0 \\ 2 - 2 \cdot 1 & 0 - 2 \cdot 0 & 0 - 2 (\frac{1}{3}) & 0 - 2 \cdot 0 \\ 2 & 0 & 0 & 0 \end{array}]$

Etapa 3.3.2.2.2

Simplifique $R_{2}$ .

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

Etapa 3.3.2.3

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

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Etapa 3.3.2.3.1

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

$[\begin{array}{c} 1 & 0 & \frac{1}{3} & 0 \\ 0 & 0 & - \frac{2}{3} & 0 \\ 2 - 2 \cdot 1 & 0 - 2 \cdot 0 & 0 - 2 (\frac{1}{3}) & 0 - 2 \cdot 0 \end{array}]$

Etapa 3.3.2.3.2

Simplifique $R_{3}$ .

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

Etapa 3.3.2.4

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

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Etapa 3.3.2.4.1

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

$[\begin{array}{c} 1 & 0 & \frac{1}{3} & 0 \\ - \frac{3}{2} \cdot 0 & - \frac{3}{2} \cdot 0 & - \frac{3}{2} (- \frac{2}{3}) & - \frac{3}{2} \cdot 0 \\ 0 & 0 & - \frac{2}{3} & 0 \end{array}]$

Etapa 3.3.2.4.2

Simplifique $R_{2}$ .

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

Etapa 3.3.2.5

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

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Etapa 3.3.2.5.1

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

$[\begin{array}{c} 1 & 0 & \frac{1}{3} & 0 \\ 0 & 0 & 1 & 0 \\ 0 + \frac{2}{3} \cdot 0 & 0 + \frac{2}{3} \cdot 0 & - \frac{2}{3} + \frac{2}{3} \cdot 1 & 0 + \frac{2}{3} \cdot 0 \end{array}]$

Etapa 3.3.2.5.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{3} & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 3.3.2.6

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

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Etapa 3.3.2.6.1

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

$[\begin{array}{c} 1 - \frac{1}{3} \cdot 0 & 0 - \frac{1}{3} \cdot 0 & \frac{1}{3} - \frac{1}{3} \cdot 1 & 0 - \frac{1}{3} \cdot 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 3.3.2.6.2

Simplifique $R_{1}$ .

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

Etapa 3.3.3

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

$x = 0$

$z = 0$

$0 = 0$

Etapa 3.3.4

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

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

Etapa 3.3.5

Write the solution as a linear combination of vectors.

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

Etapa 3.3.6

Write as a solution set.

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

Etapa 3.3.7

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

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

Etapa 4

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

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Etapa 4.1

Substitua os valores conhecidos na fórmula.

$N ([\begin{array}{c} - 3 & 0 & - 1 \\ 2 & 0 & 0 \\ 2 & 0 & 0 \end{array}] + [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Etapa 4.2

Simplifique.

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Etapa 4.2.1

Adicione os elementos correspondentes.

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

Etapa 4.2.2

Simplify each element.

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Etapa 4.2.2.1

Some $- 3$ e $1$ .

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

Etapa 4.2.2.2

Some $0$ e $0$ .

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

Etapa 4.2.2.3

Some $- 1$ e $0$ .

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

Etapa 4.2.2.4

Some $2$ e $0$ .

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

Etapa 4.2.2.5

Some $0$ e $1$ .

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

Etapa 4.2.2.6

Some $0$ e $0$ .

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

Etapa 4.2.2.7

Some $2$ e $0$ .

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

Etapa 4.2.2.8

Some $0$ e $0$ .

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

Etapa 4.2.2.9

Some $0$ e $1$ .

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

Etapa 4.3

Find the null space when $λ = - 1$ .

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Etapa 4.3.1

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

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

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 $- \frac{1}{2}$ 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 $- \frac{1}{2}$ to make the entry at $1, 1$ a $1$ .

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

Etapa 4.3.2.1.2

Simplifique $R_{1}$ .

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

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 $0$ .

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

Etapa 4.3.2.2.2

Simplifique $R_{2}$ .

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

Etapa 4.3.2.3

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

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Etapa 4.3.2.3.1

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

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

Etapa 4.3.2.3.2

Simplifique $R_{3}$ .

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

Etapa 4.3.3

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

$x + \frac{1}{2} z = 0$

$y - z = 0$

$0 = 0$

Etapa 4.3.4

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

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

Etapa 4.3.5

Write the solution as a linear combination of vectors.

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

Etapa 4.3.6

Write as a solution set.

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

Etapa 4.3.7

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

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

Etapa 5

Find the eigenvector using the eigenvalue $λ = - 2$ .

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Etapa 5.1

Substitua os valores conhecidos na fórmula.

$N ([\begin{array}{c} - 3 & 0 & - 1 \\ 2 & 0 & 0 \\ 2 & 0 & 0 \end{array}] + 2 [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

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 $2$ por cada elemento da matriz.

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

Etapa 5.2.1.2

Simplifique cada elemento da matriz.

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Etapa 5.2.1.2.1

Multiplique $2$ por $1$ .

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

Etapa 5.2.1.2.2

Multiplique $2$ por $0$ .

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

Etapa 5.2.1.2.3

Multiplique $2$ por $0$ .

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

Etapa 5.2.1.2.4

Multiplique $2$ por $0$ .

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

Etapa 5.2.1.2.5

Multiplique $2$ por $1$ .

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

Etapa 5.2.1.2.6

Multiplique $2$ por $0$ .

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

Etapa 5.2.1.2.7

Multiplique $2$ por $0$ .

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

Etapa 5.2.1.2.8

Multiplique $2$ por $0$ .

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

Etapa 5.2.1.2.9

Multiplique $2$ por $1$ .

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

Etapa 5.2.2

Adicione os elementos correspondentes.

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

Etapa 5.2.3

Simplify each element.

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Etapa 5.2.3.1

Some $- 3$ e $2$ .

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

Etapa 5.2.3.2

Some $0$ e $0$ .

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

Etapa 5.2.3.3

Some $- 1$ e $0$ .

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

Etapa 5.2.3.4

Some $2$ e $0$ .

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

Etapa 5.2.3.5

Some $0$ e $2$ .

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

Etapa 5.2.3.6

Some $0$ e $0$ .

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

Etapa 5.2.3.7

Some $2$ e $0$ .

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

Etapa 5.2.3.8

Some $0$ e $0$ .

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

Etapa 5.2.3.9

Some $0$ e $2$ .

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

Etapa 5.3

Find the null space when $λ = - 2$ .

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Etapa 5.3.1

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

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

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 $- 1$ 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 $- 1$ to make the entry at $1, 1$ a $1$ .

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

Etapa 5.3.2.1.2

Simplifique $R_{1}$ .

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

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 $0$ .

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

Etapa 5.3.2.2.2

Simplifique $R_{2}$ .

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

Etapa 5.3.2.3

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

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Etapa 5.3.2.3.1

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

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

Etapa 5.3.2.3.2

Simplifique $R_{3}$ .

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

Etapa 5.3.2.4

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

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Etapa 5.3.2.4.1

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

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

Etapa 5.3.2.4.2

Simplifique $R_{2}$ .

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

Etapa 5.3.3

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

$x + z = 0$

$y - z = 0$

$0 = 0$

Etapa 5.3.4

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

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

Etapa 5.3.5

Write the solution as a linear combination of vectors.

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

Etapa 5.3.6

Write as a solution set.

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

Etapa 5.3.7

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

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

Etapa 6

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

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