Álgebra linear Exemplos

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}]$

Etapa 1

Encontre os autovalores.

Toque para ver mais passagens...

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 $p (λ) = determinante (A - λ I_{3})$ .

Toque para ver mais passagens...

Etapa 1.3.1

Substitua $[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}]$ por $A$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] - λ I_{3})$

Etapa 1.3.2

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

$p (λ) = determinante ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Etapa 1.4

Simplifique.

Toque para ver mais passagens...

Etapa 1.4.1

Simplifique cada termo.

Toque para ver mais passagens...

Etapa 1.4.1.1

Multiplique $- λ$ por cada elemento da matriz.

$p (λ) = determinante ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2

Simplifique cada elemento da matriz.

Toque para ver mais passagens...

Etapa 1.4.1.2.1

Multiplique $- 1$ por $1$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.2

Multiplique $- λ \cdot 0$ .

Toque para ver mais passagens...

Etapa 1.4.1.2.2.1

Multiplique $0$ por $- 1$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.2.2

Multiplique $0$ por $λ$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.3

Multiplique $- λ \cdot 0$ .

Toque para ver mais passagens...

Etapa 1.4.1.2.3.1

Multiplique $0$ por $- 1$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.3.2

Multiplique $0$ por $λ$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.4

Multiplique $- λ \cdot 0$ .

Toque para ver mais passagens...

Etapa 1.4.1.2.4.1

Multiplique $0$ por $- 1$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 λ & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.4.2

Multiplique $0$ por $λ$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.5

Multiplique $- 1$ por $1$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.6

Multiplique $- λ \cdot 0$ .

Toque para ver mais passagens...

Etapa 1.4.1.2.6.1

Multiplique $0$ por $- 1$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.6.2

Multiplique $0$ por $λ$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.7

Multiplique $- λ \cdot 0$ .

Toque para ver mais passagens...

Etapa 1.4.1.2.7.1

Multiplique $0$ por $- 1$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.7.2

Multiplique $0$ por $λ$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.8

Multiplique $- λ \cdot 0$ .

Toque para ver mais passagens...

Etapa 1.4.1.2.8.1

Multiplique $0$ por $- 1$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 λ & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.8.2

Multiplique $0$ por $λ$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 1 \end{array}])$

Etapa 1.4.1.2.9

Multiplique $- 1$ por $1$ .

$p (λ) = determinante ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \end{array}])$

Etapa 1.4.2

Adicione os elementos correspondentes.

$p (λ) = determinante [\begin{array}{c} 1 - λ & - 4 + 0 & 2 + 0 \\ 2 + 0 & - 7 - λ & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 - λ \end{array}]$

Etapa 1.4.3

Simplify each element.

Toque para ver mais passagens...

Etapa 1.4.3.1

Some $- 4$ e $0$ .

$p (λ) = determinante [\begin{array}{c} 1 - λ & - 4 & 2 + 0 \\ 2 + 0 & - 7 - λ & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 - λ \end{array}]$

Etapa 1.4.3.2

Some $2$ e $0$ .

$p (λ) = determinante [\begin{array}{c} 1 - λ & - 4 & 2 \\ 2 + 0 & - 7 - λ & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 - λ \end{array}]$

Etapa 1.4.3.3

Some $2$ e $0$ .

$p (λ) = determinante [\begin{array}{c} 1 - λ & - 4 & 2 \\ 2 & - 7 - λ & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 - λ \end{array}]$

Etapa 1.4.3.4

Some $0$ e $0$ .

$p (λ) = determinante [\begin{array}{c} 1 - λ & - 4 & 2 \\ 2 & - 7 - λ & 0 \\ 3 + 0 & - 11 + 0 & 2 - λ \end{array}]$

Etapa 1.4.3.5

Some $3$ e $0$ .

$p (λ) = determinante [\begin{array}{c} 1 - λ & - 4 & 2 \\ 2 & - 7 - λ & 0 \\ 3 & - 11 + 0 & 2 - λ \end{array}]$

Etapa 1.4.3.6

Some $- 11$ e $0$ .

$p (λ) = determinante [\begin{array}{c} 1 - λ & - 4 & 2 \\ 2 & - 7 - λ & 0 \\ 3 & - 11 & 2 - λ \end{array}]$

Etapa 1.5

Find the determinant.

Toque para ver mais passagens...

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 $2$ by its cofactor and add.

Toque para ver mais passagens...

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

$| \begin{array}{c} - 4 & 2 \\ - 11 & 2 - λ \end{array} |$

Etapa 1.5.1.4

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

$- 2 | \begin{array}{c} - 4 & 2 \\ - 11 & 2 - λ \end{array} |$

Etapa 1.5.1.5

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

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

Etapa 1.5.1.6

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

$(- 7 - λ) | \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} |$

Etapa 1.5.1.7

The minor for $a_{23}$ is the determinant with row $2$ and column $3$ deleted.

$| \begin{array}{c} 1 - λ & - 4 \\ 3 & - 11 \end{array} |$

Etapa 1.5.1.8

Multiply element $a_{23}$ by its cofactor.

$0 | \begin{array}{c} 1 - λ & - 4 \\ 3 & - 11 \end{array} |$

Etapa 1.5.1.9

Add the terms together.

$p (λ) = - 2 | \begin{array}{c} - 4 & 2 \\ - 11 & 2 - λ \end{array} | + (- 7 - λ) | \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} | + 0 | \begin{array}{c} 1 - λ & - 4 \\ 3 & - 11 \end{array} |$

Etapa 1.5.2

Multiplique $0$ por $| \begin{array}{c} 1 - λ & - 4 \\ 3 & - 11 \end{array} |$ .

$p (λ) = - 2 | \begin{array}{c} - 4 & 2 \\ - 11 & 2 - λ \end{array} | + (- 7 - λ) | \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} | + 0$

Etapa 1.5.3

Avalie $| \begin{array}{c} - 4 & 2 \\ - 11 & 2 - λ \end{array} |$ .

Toque para ver mais passagens...

Etapa 1.5.3.1

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

$p (λ) = - 2 (- 4 (2 - λ) - (- 11 \cdot 2)) + (- 7 - λ) | \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} | + 0$

Etapa 1.5.3.2

Simplifique o determinante.

Toque para ver mais passagens...

Etapa 1.5.3.2.1

Simplifique cada termo.

Toque para ver mais passagens...

Etapa 1.5.3.2.1.1

Aplique a propriedade distributiva.

$p (λ) = - 2 (- 4 \cdot 2 - 4 (- λ) - (- 11 \cdot 2)) + (- 7 - λ) | \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} | + 0$

Etapa 1.5.3.2.1.2

Multiplique $- 4$ por $2$ .

$p (λ) = - 2 (- 8 - 4 (- λ) - (- 11 \cdot 2)) + (- 7 - λ) | \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} | + 0$

Etapa 1.5.3.2.1.3

Multiplique $- 1$ por $- 4$ .

$p (λ) = - 2 (- 8 + 4 λ - (- 11 \cdot 2)) + (- 7 - λ) | \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} | + 0$

Etapa 1.5.3.2.1.4

Multiplique $- (- 11 \cdot 2)$ .

Toque para ver mais passagens...

Etapa 1.5.3.2.1.4.1

Multiplique $- 11$ por $2$ .

$p (λ) = - 2 (- 8 + 4 λ - - 22) + (- 7 - λ) | \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} | + 0$

Etapa 1.5.3.2.1.4.2

Multiplique $- 1$ por $- 22$ .

$p (λ) = - 2 (- 8 + 4 λ + 22) + (- 7 - λ) | \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} | + 0$

Etapa 1.5.3.2.2

Some $- 8$ e $22$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) | \begin{array}{c} 1 - λ & 2 \\ 3 & 2 - λ \end{array} | + 0$

Etapa 1.5.4

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

Toque para ver mais passagens...

Etapa 1.5.4.1

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

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) ((1 - λ) (2 - λ) - 3 \cdot 2) + 0$

Etapa 1.5.4.2

Simplifique o determinante.

Toque para ver mais passagens...

Etapa 1.5.4.2.1

Simplifique cada termo.

Toque para ver mais passagens...

Etapa 1.5.4.2.1.1

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

Toque para ver mais passagens...

Etapa 1.5.4.2.1.1.1

Aplique a propriedade distributiva.

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (1 (2 - λ) - λ (2 - λ) - 3 \cdot 2) + 0$

Etapa 1.5.4.2.1.1.2

Aplique a propriedade distributiva.

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (1 \cdot 2 + 1 (- λ) - λ (2 - λ) - 3 \cdot 2) + 0$

Etapa 1.5.4.2.1.1.3

Aplique a propriedade distributiva.

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (1 \cdot 2 + 1 (- λ) - λ \cdot 2 - λ (- λ) - 3 \cdot 2) + 0$

Etapa 1.5.4.2.1.2

Simplifique e combine termos semelhantes.

Toque para ver mais passagens...

Etapa 1.5.4.2.1.2.1

Simplifique cada termo.

Toque para ver mais passagens...

Etapa 1.5.4.2.1.2.1.1

Multiplique $2$ por $1$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (2 + 1 (- λ) - λ \cdot 2 - λ (- λ) - 3 \cdot 2) + 0$

Etapa 1.5.4.2.1.2.1.2

Multiplique $- λ$ por $1$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (2 - λ - λ \cdot 2 - λ (- λ) - 3 \cdot 2) + 0$

Etapa 1.5.4.2.1.2.1.3

Multiplique $2$ por $- 1$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (2 - λ - 2 λ - λ (- λ) - 3 \cdot 2) + 0$

Etapa 1.5.4.2.1.2.1.4

Reescreva usando a propriedade comutativa da multiplicação.

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (2 - λ - 2 λ - 1 \cdot - 1 λ \cdot λ - 3 \cdot 2) + 0$

Etapa 1.5.4.2.1.2.1.5

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

Toque para ver mais passagens...

Etapa 1.5.4.2.1.2.1.5.1

Mova $λ$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (2 - λ - 2 λ - 1 \cdot - 1 (λ \cdot λ) - 3 \cdot 2) + 0$

Etapa 1.5.4.2.1.2.1.5.2

Multiplique $λ$ por $λ$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (2 - λ - 2 λ - 1 \cdot - 1 λ^{2} - 3 \cdot 2) + 0$

Etapa 1.5.4.2.1.2.1.6

Multiplique $- 1$ por $- 1$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (2 - λ - 2 λ + 1 λ^{2} - 3 \cdot 2) + 0$

Etapa 1.5.4.2.1.2.1.7

Multiplique $λ^{2}$ por $1$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (2 - λ - 2 λ + λ^{2} - 3 \cdot 2) + 0$

Etapa 1.5.4.2.1.2.2

Subtraia $2 λ$ de $- λ$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (2 - 3 λ + λ^{2} - 3 \cdot 2) + 0$

Etapa 1.5.4.2.1.3

Multiplique $- 3$ por $2$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (2 - 3 λ + λ^{2} - 6) + 0$

Etapa 1.5.4.2.2

Subtraia $6$ de $2$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (- 3 λ + λ^{2} - 4) + 0$

Etapa 1.5.4.2.3

Reordene $- 3 λ$ e $λ^{2}$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (λ^{2} - 3 λ - 4) + 0$

Etapa 1.5.5

Simplifique o determinante.

Toque para ver mais passagens...

Etapa 1.5.5.1

Some $- 2 (4 λ + 14) + (- 7 - λ) (λ^{2} - 3 λ - 4)$ e $0$ .

$p (λ) = - 2 (4 λ + 14) + (- 7 - λ) (λ^{2} - 3 λ - 4)$

Etapa 1.5.5.2

Simplifique cada termo.

Toque para ver mais passagens...

Etapa 1.5.5.2.1

Aplique a propriedade distributiva.

$p (λ) = - 2 (4 λ) - 2 \cdot 14 + (- 7 - λ) (λ^{2} - 3 λ - 4)$

Etapa 1.5.5.2.2

Multiplique $4$ por $- 2$ .

$p (λ) = - 8 λ - 2 \cdot 14 + (- 7 - λ) (λ^{2} - 3 λ - 4)$

Etapa 1.5.5.2.3

Multiplique $- 2$ por $14$ .

$p (λ) = - 8 λ - 28 + (- 7 - λ) (λ^{2} - 3 λ - 4)$

Etapa 1.5.5.2.4

Expanda $(- 7 - λ) (λ^{2} - 3 λ - 4)$ multiplicando cada termo na primeira expressão por cada um dos termos na segunda expressão.

$p (λ) = - 8 λ - 28 - 7 λ^{2} - 7 (- 3 λ) - 7 \cdot - 4 - λ \cdot λ^{2} - λ (- 3 λ) - λ \cdot - 4$

Etapa 1.5.5.2.5

Simplifique cada termo.

Toque para ver mais passagens...

Etapa 1.5.5.2.5.1

Multiplique $- 3$ por $- 7$ .

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ - 7 \cdot - 4 - λ \cdot λ^{2} - λ (- 3 λ) - λ \cdot - 4$

Etapa 1.5.5.2.5.2

Multiplique $- 7$ por $- 4$ .

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ + 28 - λ \cdot λ^{2} - λ (- 3 λ) - λ \cdot - 4$

Etapa 1.5.5.2.5.3

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

Toque para ver mais passagens...

Etapa 1.5.5.2.5.3.1

Mova $λ^{2}$ .

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ + 28 - (λ^{2} λ) - λ (- 3 λ) - λ \cdot - 4$

Etapa 1.5.5.2.5.3.2

Multiplique $λ^{2}$ por $λ$ .

Toque para ver mais passagens...

Etapa 1.5.5.2.5.3.2.1

Eleve $λ$ à potência de $1$ .

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ + 28 - (λ^{2} λ^{1}) - λ (- 3 λ) - λ \cdot - 4$

Etapa 1.5.5.2.5.3.2.2

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

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ + 28 - λ^{2 + 1} - λ (- 3 λ) - λ \cdot - 4$

Etapa 1.5.5.2.5.3.3

Some $2$ e $1$ .

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ + 28 - λ^{3} - λ (- 3 λ) - λ \cdot - 4$

Etapa 1.5.5.2.5.4

Reescreva usando a propriedade comutativa da multiplicação.

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ + 28 - λ^{3} - 1 \cdot - 3 λ \cdot λ - λ \cdot - 4$

Etapa 1.5.5.2.5.5

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

Toque para ver mais passagens...

Etapa 1.5.5.2.5.5.1

Mova $λ$ .

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ + 28 - λ^{3} - 1 \cdot - 3 (λ \cdot λ) - λ \cdot - 4$

Etapa 1.5.5.2.5.5.2

Multiplique $λ$ por $λ$ .

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ + 28 - λ^{3} - 1 \cdot - 3 λ^{2} - λ \cdot - 4$

Etapa 1.5.5.2.5.6

Multiplique $- 1$ por $- 3$ .

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ + 28 - λ^{3} + 3 λ^{2} - λ \cdot - 4$

Etapa 1.5.5.2.5.7

Multiplique $- 4$ por $- 1$ .

$p (λ) = - 8 λ - 28 - 7 λ^{2} + 21 λ + 28 - λ^{3} + 3 λ^{2} + 4 λ$

Etapa 1.5.5.2.6

Some $- 7 λ^{2}$ e $3 λ^{2}$ .

$p (λ) = - 8 λ - 28 - 4 λ^{2} + 21 λ + 28 - λ^{3} + 4 λ$

Etapa 1.5.5.2.7

Some $21 λ$ e $4 λ$ .

$p (λ) = - 8 λ - 28 - 4 λ^{2} + 25 λ + 28 - λ^{3}$

Etapa 1.5.5.3

Combine os termos opostos em $- 8 λ - 28 - 4 λ^{2} + 25 λ + 28 - λ^{3}$ .

Toque para ver mais passagens...

Etapa 1.5.5.3.1

Some $- 28$ e $28$ .

$p (λ) = - 8 λ - 4 λ^{2} + 25 λ + 0 - λ^{3}$

Etapa 1.5.5.3.2

Some $- 8 λ - 4 λ^{2} + 25 λ$ e $0$ .

$p (λ) = - 8 λ - 4 λ^{2} + 25 λ - λ^{3}$

Etapa 1.5.5.4

Some $- 8 λ$ e $25 λ$ .

$p (λ) = - 4 λ^{2} + 17 λ - λ^{3}$

Etapa 1.5.5.5

Mova $17 λ$ .

$p (λ) = - 4 λ^{2} - λ^{3} + 17 λ$

Etapa 1.5.5.6

Reordene $- 4 λ^{2}$ e $- λ^{3}$ .

$p (λ) = - λ^{3} - 4 λ^{2} + 17 λ$

Etapa 1.6

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

$- λ^{3} - 4 λ^{2} + 17 λ = 0$

Etapa 1.7

Resolva $λ$ .

Toque para ver mais passagens...

Etapa 1.7.1

Fatore $- λ$ de $- λ^{3} - 4 λ^{2} + 17 λ$ .

Toque para ver mais passagens...

Etapa 1.7.1.1

Fatore $- λ$ de $- λ^{3}$ .

$- λ \cdot λ^{2} - 4 λ^{2} + 17 λ = 0$

Etapa 1.7.1.2

Fatore $- λ$ de $- 4 λ^{2}$ .

$- λ \cdot λ^{2} - λ (4 λ) + 17 λ = 0$

Etapa 1.7.1.3

Fatore $- λ$ de $17 λ$ .

$- λ \cdot λ^{2} - λ (4 λ) - λ \cdot - 17 = 0$

Etapa 1.7.1.4

Fatore $- λ$ de $- λ (λ^{2}) - λ (4 λ)$ .

$- λ (λ^{2} + 4 λ) - λ \cdot - 17 = 0$

Etapa 1.7.1.5

Fatore $- λ$ de $- λ (λ^{2} + 4 λ) - λ (- 17)$ .

$- λ (λ^{2} + 4 λ - 17) = 0$

Etapa 1.7.2

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

$λ = 0$

$λ^{2} + 4 λ - 17 = 0$

Etapa 1.7.3

Defina $λ$ como igual a $0$ .

$λ = 0$

Etapa 1.7.4

Defina $λ^{2} + 4 λ - 17$ como igual a $0$ e resolva para $λ$ .

Toque para ver mais passagens...

Etapa 1.7.4.1

Defina $λ^{2} + 4 λ - 17$ como igual a $0$ .

$λ^{2} + 4 λ - 17 = 0$

Etapa 1.7.4.2

Resolva $λ^{2} + 4 λ - 17 = 0$ para $λ$ .

Toque para ver mais passagens...

Etapa 1.7.4.2.1

Use a fórmula quadrática para encontrar as soluções.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Etapa 1.7.4.2.2

Substitua os valores $a = 1$ , $b = 4$ e $c = - 17$ na fórmula quadrática e resolva $λ$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot - 17)}}{2 \cdot 1}$

Etapa 1.7.4.2.3

Simplifique.

Toque para ver mais passagens...

Etapa 1.7.4.2.3.1

Simplifique o numerador.

Toque para ver mais passagens...

Etapa 1.7.4.2.3.1.1

Eleve $4$ à potência de $2$ .

$λ = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 17}}{2 \cdot 1}$

Etapa 1.7.4.2.3.1.2

Multiplique $- 4 \cdot 1 \cdot - 17$ .

Toque para ver mais passagens...

Etapa 1.7.4.2.3.1.2.1

Multiplique $- 4$ por $1$ .

$λ = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 17}}{2 \cdot 1}$

Etapa 1.7.4.2.3.1.2.2

Multiplique $- 4$ por $- 17$ .

$λ = \frac{- 4 \pm \sqrt{16 + 68}}{2 \cdot 1}$

Etapa 1.7.4.2.3.1.3

Some $16$ e $68$ .

$λ = \frac{- 4 \pm \sqrt{84}}{2 \cdot 1}$

Etapa 1.7.4.2.3.1.4

Reescreva $84$ como $2^{2} \cdot 21$ .

Toque para ver mais passagens...

Etapa 1.7.4.2.3.1.4.1

Fatore $4$ de $84$ .

$λ = \frac{- 4 \pm \sqrt{4 (21)}}{2 \cdot 1}$

Etapa 1.7.4.2.3.1.4.2

Reescreva $4$ como $2^{2}$ .

$λ = \frac{- 4 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 1}$

Etapa 1.7.4.2.3.1.5

Elimine os termos abaixo do radical.

$λ = \frac{- 4 \pm 2 \sqrt{21}}{2 \cdot 1}$

Etapa 1.7.4.2.3.2

Multiplique $2$ por $1$ .

$λ = \frac{- 4 \pm 2 \sqrt{21}}{2}$

Etapa 1.7.4.2.3.3

Simplifique $\frac{- 4 \pm 2 \sqrt{21}}{2}$ .

$λ = - 2 \pm \sqrt{21}$

Etapa 1.7.4.2.4

A resposta final é a combinação das duas soluções.

$λ = - 2 + \sqrt{21}, - 2 - \sqrt{21}$

Etapa 1.7.5

A solução final são todos os valores que tornam $- λ (λ^{2} + 4 λ - 17) = 0$ verdadeiro.

$λ = 0, - 2 + \sqrt{21}, - 2 - \sqrt{21}$

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 $λ = 0$ .

Toque para ver mais passagens...

Etapa 3.1

Substitua os valores conhecidos na fórmula.

$N ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + 0 [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Etapa 3.2

Simplifique.

Toque para ver mais passagens...

Etapa 3.2.1

Simplifique cada termo.

Toque para ver mais passagens...

Etapa 3.2.1.1

Multiplique $0$ por cada elemento da matriz.

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 0 \cdot 1 & 0 \cdot 0 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 1 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Etapa 3.2.1.2

Simplifique cada elemento da matriz.

Toque para ver mais passagens...

Etapa 3.2.1.2.1

Multiplique $0$ por $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 0 & 0 \cdot 0 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 1 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Etapa 3.2.1.2.2

Multiplique $0$ por $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 1 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Etapa 3.2.1.2.3

Multiplique $0$ por $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 \cdot 0 & 0 \cdot 1 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Etapa 3.2.1.2.4

Multiplique $0$ por $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 \cdot 1 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Etapa 3.2.1.2.5

Multiplique $0$ por $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Etapa 3.2.1.2.6

Multiplique $0$ por $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Etapa 3.2.1.2.7

Multiplique $0$ por $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Etapa 3.2.1.2.8

Multiplique $0$ por $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \cdot 1 \end{array}]$

Etapa 3.2.1.2.9

Multiplique $0$ por $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}]$

Etapa 3.2.2

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

Toque para ver mais passagens...

Etapa 3.2.2.1

Adicione os elementos correspondentes.

$[\begin{array}{c} 1 + 0 & - 4 + 0 & 2 + 0 \\ 2 + 0 & - 7 + 0 & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 0 \end{array}]$

Etapa 3.2.2.2

Simplify each element.

Toque para ver mais passagens...

Etapa 3.2.2.2.1

Some $1$ e $0$ .

$[\begin{array}{c} 1 & - 4 + 0 & 2 + 0 \\ 2 + 0 & - 7 + 0 & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 0 \end{array}]$

Etapa 3.2.2.2.2

Some $- 4$ e $0$ .

$[\begin{array}{c} 1 & - 4 & 2 + 0 \\ 2 + 0 & - 7 + 0 & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 0 \end{array}]$

Etapa 3.2.2.2.3

Some $2$ e $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 + 0 & - 7 + 0 & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 0 \end{array}]$

Etapa 3.2.2.2.4

Some $2$ e $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 + 0 & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 0 \end{array}]$

Etapa 3.2.2.2.5

Some $- 7$ e $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 0 \end{array}]$

Etapa 3.2.2.2.6

Some $0$ e $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 + 0 & - 11 + 0 & 2 + 0 \end{array}]$

Etapa 3.2.2.2.7

Some $3$ e $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 + 0 & 2 + 0 \end{array}]$

Etapa 3.2.2.2.8

Some $- 11$ e $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 + 0 \end{array}]$

Etapa 3.2.2.2.9

Some $2$ e $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}]$

Etapa 3.3

Find the null space when $λ = 0$ .

Toque para ver mais passagens...

Etapa 3.3.1

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

$[\begin{array}{c} 1 & - 4 & 2 & 0 \\ 2 & - 7 & 0 & 0 \\ 3 & - 11 & 2 & 0 \end{array}]$

Etapa 3.3.2

Encontre a forma escalonada reduzida por linhas.

Toque para ver mais passagens...

Etapa 3.3.2.1

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

Toque para ver mais passagens...

Etapa 3.3.2.1.1

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

$[\begin{array}{c} 1 & - 4 & 2 & 0 \\ 2 - 2 \cdot 1 & - 7 - 2 \cdot - 4 & 0 - 2 \cdot 2 & 0 - 2 \cdot 0 \\ 3 & - 11 & 2 & 0 \end{array}]$

Etapa 3.3.2.1.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & - 4 & 2 & 0 \\ 0 & 1 & - 4 & 0 \\ 3 & - 11 & 2 & 0 \end{array}]$

Etapa 3.3.2.2

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

Toque para ver mais passagens...

Etapa 3.3.2.2.1

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

$[\begin{array}{c} 1 & - 4 & 2 & 0 \\ 0 & 1 & - 4 & 0 \\ 3 - 3 \cdot 1 & - 11 - 3 \cdot - 4 & 2 - 3 \cdot 2 & 0 - 3 \cdot 0 \end{array}]$

Etapa 3.3.2.2.2

Simplifique $R_{3}$ .

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

Etapa 3.3.2.3

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

Toque para ver mais passagens...

Etapa 3.3.2.3.1

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

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

Etapa 3.3.2.3.2

Simplifique $R_{3}$ .

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

Etapa 3.3.2.4

Perform the row operation $R_{1} = R_{1} + 4 R_{2}$ to make the entry at $1, 2$ a $0$ .

Toque para ver mais passagens...

Etapa 3.3.2.4.1

Perform the row operation $R_{1} = R_{1} + 4 R_{2}$ to make the entry at $1, 2$ a $0$ .

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

Etapa 3.3.2.4.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - 14 & 0 \\ 0 & 1 & - 4 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 3.3.3

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

$x - 14 z = 0$

$y - 4 z = 0$

$0 = 0$

Etapa 3.3.4

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

$[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 14 z \\ 4 z \\ z \end{array}]$

Etapa 3.3.5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x \\ y \\ z \end{array}] = z [\begin{array}{c} 14 \\ 4 \\ 1 \end{array}]$

Etapa 3.3.6

Write as a solution set.

${z [\begin{array}{c} 14 \\ 4 \\ 1 \end{array}] | z \in R}$

Etapa 3.3.7

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

${[\begin{array}{c} 14 \\ 4 \\ 1 \end{array}]}$

Etapa 4

Find the eigenvector using the eigenvalue $λ = - 2 + \sqrt{21}$ .

Toque para ver mais passagens...

Etapa 4.1

Substitua os valores conhecidos na fórmula.

$N ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] - (- 2 + \sqrt{21}) [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Etapa 4.2

Simplifique.

Toque para ver mais passagens...

Etapa 4.2.1

Simplifique cada termo.

Toque para ver mais passagens...

Etapa 4.2.1.1

Aplique a propriedade distributiva.

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + (- - 2 - \sqrt{21}) [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Etapa 4.2.1.2

Multiplique $- 1$ por $- 2$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + (2 - \sqrt{21}) [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Etapa 4.2.1.3

Multiplique $2 - \sqrt{21}$ por cada elemento da matriz.

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} (2 - \sqrt{21}) \cdot 1 & (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 & (2 - \sqrt{21}) \cdot 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 \end{array}]$

Etapa 4.2.1.4

Simplifique cada elemento da matriz.

Toque para ver mais passagens...

Etapa 4.2.1.4.1

Multiplique $2 - \sqrt{21}$ por $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 - \sqrt{21} & (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 & (2 - \sqrt{21}) \cdot 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 \end{array}]$

Etapa 4.2.1.4.2

Multiplique $2 - \sqrt{21}$ por $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 - \sqrt{21} & 0 & (2 - \sqrt{21}) \cdot 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 & (2 - \sqrt{21}) \cdot 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 \end{array}]$

Etapa 4.2.1.4.3

Multiplique $2 - \sqrt{21}$ por $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 - \sqrt{21} & 0 & 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 & (2 - \sqrt{21}) \cdot 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 \end{array}]$

Etapa 4.2.1.4.4

Multiplique $2 - \sqrt{21}$ por $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 - \sqrt{21} & 0 & 0 \\ 0 & (2 - \sqrt{21}) \cdot 1 & (2 - \sqrt{21}) \cdot 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 \end{array}]$

Etapa 4.2.1.4.5

Multiplique $2 - \sqrt{21}$ por $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 - \sqrt{21} & 0 & 0 \\ 0 & 2 - \sqrt{21} & (2 - \sqrt{21}) \cdot 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 \end{array}]$

Etapa 4.2.1.4.6

Multiplique $2 - \sqrt{21}$ por $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 - \sqrt{21} & 0 & 0 \\ 0 & 2 - \sqrt{21} & 0 \\ (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 \end{array}]$

Etapa 4.2.1.4.7

Multiplique $2 - \sqrt{21}$ por $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 - \sqrt{21} & 0 & 0 \\ 0 & 2 - \sqrt{21} & 0 \\ 0 & (2 - \sqrt{21}) \cdot 0 & (2 - \sqrt{21}) \cdot 1 \end{array}]$

Etapa 4.2.1.4.8

Multiplique $2 - \sqrt{21}$ por $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 - \sqrt{21} & 0 & 0 \\ 0 & 2 - \sqrt{21} & 0 \\ 0 & 0 & (2 - \sqrt{21}) \cdot 1 \end{array}]$

Etapa 4.2.1.4.9

Multiplique $2 - \sqrt{21}$ por $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 - \sqrt{21} & 0 & 0 \\ 0 & 2 - \sqrt{21} & 0 \\ 0 & 0 & 2 - \sqrt{21} \end{array}]$

Etapa 4.2.2

Adicione os elementos correspondentes.

$[\begin{array}{c} 1 + 2 - \sqrt{21} & - 4 + 0 & 2 + 0 \\ 2 + 0 & - 7 + 2 - \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 - \sqrt{21} \end{array}]$

Etapa 4.2.3

Simplify each element.

Toque para ver mais passagens...

Etapa 4.2.3.1

Some $1$ e $2$ .

$[\begin{array}{c} 3 - \sqrt{21} & - 4 + 0 & 2 + 0 \\ 2 + 0 & - 7 + 2 - \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 - \sqrt{21} \end{array}]$

Etapa 4.2.3.2

Some $- 4$ e $0$ .

$[\begin{array}{c} 3 - \sqrt{21} & - 4 & 2 + 0 \\ 2 + 0 & - 7 + 2 - \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 - \sqrt{21} \end{array}]$

Etapa 4.2.3.3

Some $2$ e $0$ .

$[\begin{array}{c} 3 - \sqrt{21} & - 4 & 2 \\ 2 + 0 & - 7 + 2 - \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 - \sqrt{21} \end{array}]$

Etapa 4.2.3.4

Some $2$ e $0$ .

$[\begin{array}{c} 3 - \sqrt{21} & - 4 & 2 \\ 2 & - 7 + 2 - \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 - \sqrt{21} \end{array}]$

Etapa 4.2.3.5

Some $- 7$ e $2$ .

$[\begin{array}{c} 3 - \sqrt{21} & - 4 & 2 \\ 2 & - 5 - \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 - \sqrt{21} \end{array}]$

Etapa 4.2.3.6

Some $0$ e $0$ .

$[\begin{array}{c} 3 - \sqrt{21} & - 4 & 2 \\ 2 & - 5 - \sqrt{21} & 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 - \sqrt{21} \end{array}]$

Etapa 4.2.3.7

Some $3$ e $0$ .

$[\begin{array}{c} 3 - \sqrt{21} & - 4 & 2 \\ 2 & - 5 - \sqrt{21} & 0 \\ 3 & - 11 + 0 & 2 + 2 - \sqrt{21} \end{array}]$

Etapa 4.2.3.8

Some $- 11$ e $0$ .

$[\begin{array}{c} 3 - \sqrt{21} & - 4 & 2 \\ 2 & - 5 - \sqrt{21} & 0 \\ 3 & - 11 & 2 + 2 - \sqrt{21} \end{array}]$

Etapa 4.2.3.9

Some $2$ e $2$ .

$[\begin{array}{c} 3 - \sqrt{21} & - 4 & 2 \\ 2 & - 5 - \sqrt{21} & 0 \\ 3 & - 11 & 4 - \sqrt{21} \end{array}]$

Etapa 4.3

Find the null space when $λ = - 2 + \sqrt{21}$ .

Toque para ver mais passagens...

Etapa 4.3.1

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

$[\begin{array}{c} 3 - \sqrt{21} & - 4 & 2 & 0 \\ 2 & - 5 - \sqrt{21} & 0 & 0 \\ 3 & - 11 & 4 - \sqrt{21} & 0 \end{array}]$

Etapa 4.3.2

Encontre a forma escalonada reduzida por linhas.

Toque para ver mais passagens...

Etapa 4.3.2.1

Multiply each element of $R_{1}$ by $\frac{1}{3 - \sqrt{21}}$ to make the entry at $1, 1$ a $1$ .

Toque para ver mais passagens...

Etapa 4.3.2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{3 - \sqrt{21}}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{3 - \sqrt{21}}{3 - \sqrt{21}} & \frac{- 4}{3 - \sqrt{21}} & \frac{2}{3 - \sqrt{21}} & \frac{0}{3 - \sqrt{21}} \\ 2 & - 5 - \sqrt{21} & 0 & 0 \\ 3 & - 11 & 4 - \sqrt{21} & 0 \end{array}]$

Etapa 4.3.2.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & \frac{3 + \sqrt{21}}{3} & - \frac{3 + \sqrt{21}}{6} & 0 \\ 2 & - 5 - \sqrt{21} & 0 & 0 \\ 3 & - 11 & 4 - \sqrt{21} & 0 \end{array}]$

Etapa 4.3.2.2

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

Toque para ver mais passagens...

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 $0$ .

$[\begin{array}{c} 1 & \frac{3 + \sqrt{21}}{3} & - \frac{3 + \sqrt{21}}{6} & 0 \\ 2 - 2 \cdot 1 & - 5 - \sqrt{21} - 2 \frac{3 + \sqrt{21}}{3} & 0 - 2 (- \frac{3 + \sqrt{21}}{6}) & 0 - 2 \cdot 0 \\ 3 & - 11 & 4 - \sqrt{21} & 0 \end{array}]$

Etapa 4.3.2.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3 + \sqrt{21}}{3} & - \frac{3 + \sqrt{21}}{6} & 0 \\ 0 & - \frac{21 + 5 \sqrt{21}}{3} & \frac{3 + \sqrt{21}}{3} & 0 \\ 3 & - 11 & 4 - \sqrt{21} & 0 \end{array}]$

Etapa 4.3.2.3

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

Toque para ver mais passagens...

Etapa 4.3.2.3.1

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

$[\begin{array}{c} 1 & \frac{3 + \sqrt{21}}{3} & - \frac{3 + \sqrt{21}}{6} & 0 \\ 0 & - \frac{21 + 5 \sqrt{21}}{3} & \frac{3 + \sqrt{21}}{3} & 0 \\ 3 - 3 \cdot 1 & - 11 - 3 \frac{3 + \sqrt{21}}{3} & 4 - \sqrt{21} - 3 (- \frac{3 + \sqrt{21}}{6}) & 0 - 3 \cdot 0 \end{array}]$

Etapa 4.3.2.3.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{3 + \sqrt{21}}{3} & - \frac{3 + \sqrt{21}}{6} & 0 \\ 0 & - \frac{21 + 5 \sqrt{21}}{3} & \frac{3 + \sqrt{21}}{3} & 0 \\ 0 & - 14 - \sqrt{21} & \frac{11 - \sqrt{21}}{2} & 0 \end{array}]$

Etapa 4.3.2.4

Multiply each element of $R_{2}$ by $- \frac{3}{21 + 5 \sqrt{21}}$ to make the entry at $2, 2$ a $1$ .

Toque para ver mais passagens...

Etapa 4.3.2.4.1

Multiply each element of $R_{2}$ by $- \frac{3}{21 + 5 \sqrt{21}}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{3 + \sqrt{21}}{3} & - \frac{3 + \sqrt{21}}{6} & 0 \\ - \frac{3}{21 + 5 \sqrt{21}} \cdot 0 & - \frac{3}{21 + 5 \sqrt{21}} (- \frac{21 + 5 \sqrt{21}}{3}) & - \frac{3}{21 + 5 \sqrt{21}} \cdot \frac{3 + \sqrt{21}}{3} & - \frac{3}{21 + 5 \sqrt{21}} \cdot 0 \\ 0 & - 14 - \sqrt{21} & \frac{11 - \sqrt{21}}{2} & 0 \end{array}]$

Etapa 4.3.2.4.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3 + \sqrt{21}}{3} & - \frac{3 + \sqrt{21}}{6} & 0 \\ 0 & 1 & - \frac{7 - \sqrt{21}}{14} & 0 \\ 0 & - 14 - \sqrt{21} & \frac{11 - \sqrt{21}}{2} & 0 \end{array}]$

Etapa 4.3.2.5

Perform the row operation $R_{3} = R_{3} - (- 14 - \sqrt{21}) R_{2}$ to make the entry at $3, 2$ a $0$ .

Toque para ver mais passagens...

Etapa 4.3.2.5.1

Perform the row operation $R_{3} = R_{3} - (- 14 - \sqrt{21}) R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{3 + \sqrt{21}}{3} & - \frac{3 + \sqrt{21}}{6} & 0 \\ 0 & 1 & - \frac{7 - \sqrt{21}}{14} & 0 \\ 0 - (- 14 - \sqrt{21}) \cdot 0 & - 14 - \sqrt{21} - (- 14 - \sqrt{21}) \cdot 1 & \frac{11 - \sqrt{21}}{2} - (- 14 - \sqrt{21}) (- \frac{7 - \sqrt{21}}{14}) & 0 - (- 14 - \sqrt{21}) \cdot 0 \end{array}]$

Etapa 4.3.2.5.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{3 + \sqrt{21}}{3} & - \frac{3 + \sqrt{21}}{6} & 0 \\ 0 & 1 & - \frac{7 - \sqrt{21}}{14} & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 4.3.2.6

Perform the row operation $R_{1} = R_{1} - \frac{3 + \sqrt{21}}{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

Toque para ver mais passagens...

Etapa 4.3.2.6.1

Perform the row operation $R_{1} = R_{1} - \frac{3 + \sqrt{21}}{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{3 + \sqrt{21}}{3} \cdot 0 & \frac{3 + \sqrt{21}}{3} - \frac{3 + \sqrt{21}}{3} \cdot 1 & - \frac{3 + \sqrt{21}}{6} - \frac{3 + \sqrt{21}}{3} (- \frac{7 - \sqrt{21}}{14}) & 0 - \frac{3 + \sqrt{21}}{3} \cdot 0 \\ 0 & 1 & - \frac{7 - \sqrt{21}}{14} & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 4.3.2.6.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - \frac{7 + \sqrt{21}}{14} & 0 \\ 0 & 1 & - \frac{7 - \sqrt{21}}{14} & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 4.3.3

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

$x - \frac{7 + \sqrt{21}}{14} z = 0$

$y - \frac{7 - \sqrt{21}}{14} z = 0$

$0 = 0$

Etapa 4.3.4

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

$[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} \frac{z}{2} + \frac{z \sqrt{21}}{14} \\ \frac{z}{2} - \frac{z \sqrt{21}}{14} \\ z \end{array}]$

Etapa 4.3.5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x \\ y \\ z \end{array}] = z [\begin{array}{c} \frac{1}{2} + \frac{\sqrt{21}}{14} \\ \frac{1}{2} - \frac{\sqrt{21}}{14} \\ 1 \end{array}]$

Etapa 4.3.6

Write as a solution set.

${z [\begin{array}{c} \frac{1}{2} + \frac{\sqrt{21}}{14} \\ \frac{1}{2} - \frac{\sqrt{21}}{14} \\ 1 \end{array}] | z \in R}$

Etapa 4.3.7

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

${[\begin{array}{c} \frac{1}{2} + \frac{\sqrt{21}}{14} \\ \frac{1}{2} - \frac{\sqrt{21}}{14} \\ 1 \end{array}]}$

Etapa 5

Find the eigenvector using the eigenvalue $λ = - 2 - \sqrt{21}$ .

Toque para ver mais passagens...

Etapa 5.1

Substitua os valores conhecidos na fórmula.

$N ([\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] - (- 2 - \sqrt{21}) [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Etapa 5.2

Simplifique.

Toque para ver mais passagens...

Etapa 5.2.1

Simplifique cada termo.

Toque para ver mais passagens...

Etapa 5.2.1.1

Aplique a propriedade distributiva.

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + (- - 2 - - \sqrt{21}) [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Etapa 5.2.1.2

Multiplique $- 1$ por $- 2$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + (2 - - \sqrt{21}) [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Etapa 5.2.1.3

Multiplique $- - \sqrt{21}$ .

Toque para ver mais passagens...

Etapa 5.2.1.3.1

Multiplique $- 1$ por $- 1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + (2 + 1 \sqrt{21}) [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Etapa 5.2.1.3.2

Multiplique $\sqrt{21}$ por $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + (2 + \sqrt{21}) [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Etapa 5.2.1.4

Multiplique $2 + \sqrt{21}$ por cada elemento da matriz.

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} (2 + \sqrt{21}) \cdot 1 & (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 & (2 + \sqrt{21}) \cdot 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 \end{array}]$

Etapa 5.2.1.5

Simplifique cada elemento da matriz.

Toque para ver mais passagens...

Etapa 5.2.1.5.1

Multiplique $2 + \sqrt{21}$ por $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 + \sqrt{21} & (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 & (2 + \sqrt{21}) \cdot 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 \end{array}]$

Etapa 5.2.1.5.2

Multiplique $2 + \sqrt{21}$ por $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 + \sqrt{21} & 0 & (2 + \sqrt{21}) \cdot 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 & (2 + \sqrt{21}) \cdot 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 \end{array}]$

Etapa 5.2.1.5.3

Multiplique $2 + \sqrt{21}$ por $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 + \sqrt{21} & 0 & 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 & (2 + \sqrt{21}) \cdot 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 \end{array}]$

Etapa 5.2.1.5.4

Multiplique $2 + \sqrt{21}$ por $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 + \sqrt{21} & 0 & 0 \\ 0 & (2 + \sqrt{21}) \cdot 1 & (2 + \sqrt{21}) \cdot 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 \end{array}]$

Etapa 5.2.1.5.5

Multiplique $2 + \sqrt{21}$ por $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 + \sqrt{21} & 0 & 0 \\ 0 & 2 + \sqrt{21} & (2 + \sqrt{21}) \cdot 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 \end{array}]$

Etapa 5.2.1.5.6

Multiplique $2 + \sqrt{21}$ por $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 + \sqrt{21} & 0 & 0 \\ 0 & 2 + \sqrt{21} & 0 \\ (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 \end{array}]$

Etapa 5.2.1.5.7

Multiplique $2 + \sqrt{21}$ por $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 + \sqrt{21} & 0 & 0 \\ 0 & 2 + \sqrt{21} & 0 \\ 0 & (2 + \sqrt{21}) \cdot 0 & (2 + \sqrt{21}) \cdot 1 \end{array}]$

Etapa 5.2.1.5.8

Multiplique $2 + \sqrt{21}$ por $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 + \sqrt{21} & 0 & 0 \\ 0 & 2 + \sqrt{21} & 0 \\ 0 & 0 & (2 + \sqrt{21}) \cdot 1 \end{array}]$

Etapa 5.2.1.5.9

Multiplique $2 + \sqrt{21}$ por $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 2 & - 7 & 0 \\ 3 & - 11 & 2 \end{array}] + [\begin{array}{c} 2 + \sqrt{21} & 0 & 0 \\ 0 & 2 + \sqrt{21} & 0 \\ 0 & 0 & 2 + \sqrt{21} \end{array}]$

Etapa 5.2.2

Adicione os elementos correspondentes.

$[\begin{array}{c} 1 + 2 + \sqrt{21} & - 4 + 0 & 2 + 0 \\ 2 + 0 & - 7 + 2 + \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 + \sqrt{21} \end{array}]$

Etapa 5.2.3

Simplify each element.

Toque para ver mais passagens...

Etapa 5.2.3.1

Some $1$ e $2$ .

$[\begin{array}{c} 3 + \sqrt{21} & - 4 + 0 & 2 + 0 \\ 2 + 0 & - 7 + 2 + \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 + \sqrt{21} \end{array}]$

Etapa 5.2.3.2

Some $- 4$ e $0$ .

$[\begin{array}{c} 3 + \sqrt{21} & - 4 & 2 + 0 \\ 2 + 0 & - 7 + 2 + \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 + \sqrt{21} \end{array}]$

Etapa 5.2.3.3

Some $2$ e $0$ .

$[\begin{array}{c} 3 + \sqrt{21} & - 4 & 2 \\ 2 + 0 & - 7 + 2 + \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 + \sqrt{21} \end{array}]$

Etapa 5.2.3.4

Some $2$ e $0$ .

$[\begin{array}{c} 3 + \sqrt{21} & - 4 & 2 \\ 2 & - 7 + 2 + \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 + \sqrt{21} \end{array}]$

Etapa 5.2.3.5

Some $- 7$ e $2$ .

$[\begin{array}{c} 3 + \sqrt{21} & - 4 & 2 \\ 2 & - 5 + \sqrt{21} & 0 + 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 + \sqrt{21} \end{array}]$

Etapa 5.2.3.6

Some $0$ e $0$ .

$[\begin{array}{c} 3 + \sqrt{21} & - 4 & 2 \\ 2 & - 5 + \sqrt{21} & 0 \\ 3 + 0 & - 11 + 0 & 2 + 2 + \sqrt{21} \end{array}]$

Etapa 5.2.3.7

Some $3$ e $0$ .

$[\begin{array}{c} 3 + \sqrt{21} & - 4 & 2 \\ 2 & - 5 + \sqrt{21} & 0 \\ 3 & - 11 + 0 & 2 + 2 + \sqrt{21} \end{array}]$

Etapa 5.2.3.8

Some $- 11$ e $0$ .

$[\begin{array}{c} 3 + \sqrt{21} & - 4 & 2 \\ 2 & - 5 + \sqrt{21} & 0 \\ 3 & - 11 & 2 + 2 + \sqrt{21} \end{array}]$

Etapa 5.2.3.9

Some $2$ e $2$ .

$[\begin{array}{c} 3 + \sqrt{21} & - 4 & 2 \\ 2 & - 5 + \sqrt{21} & 0 \\ 3 & - 11 & 4 + \sqrt{21} \end{array}]$

Etapa 5.3

Find the null space when $λ = - 2 - \sqrt{21}$ .

Toque para ver mais passagens...

Etapa 5.3.1

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

$[\begin{array}{c} 3 + \sqrt{21} & - 4 & 2 & 0 \\ 2 & - 5 + \sqrt{21} & 0 & 0 \\ 3 & - 11 & 4 + \sqrt{21} & 0 \end{array}]$

Etapa 5.3.2

Encontre a forma escalonada reduzida por linhas.

Toque para ver mais passagens...

Etapa 5.3.2.1

Multiply each element of $R_{1}$ by $\frac{1}{3 + \sqrt{21}}$ to make the entry at $1, 1$ a $1$ .

Toque para ver mais passagens...

Etapa 5.3.2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{3 + \sqrt{21}}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{3 + \sqrt{21}}{3 + \sqrt{21}} & \frac{- 4}{3 + \sqrt{21}} & \frac{2}{3 + \sqrt{21}} & \frac{0}{3 + \sqrt{21}} \\ 2 & - 5 + \sqrt{21} & 0 & 0 \\ 3 & - 11 & 4 + \sqrt{21} & 0 \end{array}]$

Etapa 5.3.2.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & \frac{3 - \sqrt{21}}{3} & - \frac{3 - \sqrt{21}}{6} & 0 \\ 2 & - 5 + \sqrt{21} & 0 & 0 \\ 3 & - 11 & 4 + \sqrt{21} & 0 \end{array}]$

Etapa 5.3.2.2

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

Toque para ver mais passagens...

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 $0$ .

$[\begin{array}{c} 1 & \frac{3 - \sqrt{21}}{3} & - \frac{3 - \sqrt{21}}{6} & 0 \\ 2 - 2 \cdot 1 & - 5 + \sqrt{21} - 2 \frac{3 - \sqrt{21}}{3} & 0 - 2 (- \frac{3 - \sqrt{21}}{6}) & 0 - 2 \cdot 0 \\ 3 & - 11 & 4 + \sqrt{21} & 0 \end{array}]$

Etapa 5.3.2.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3 - \sqrt{21}}{3} & - \frac{3 - \sqrt{21}}{6} & 0 \\ 0 & - \frac{21 - 5 \sqrt{21}}{3} & \frac{3 - \sqrt{21}}{3} & 0 \\ 3 & - 11 & 4 + \sqrt{21} & 0 \end{array}]$

Etapa 5.3.2.3

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

Toque para ver mais passagens...

Etapa 5.3.2.3.1

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

$[\begin{array}{c} 1 & \frac{3 - \sqrt{21}}{3} & - \frac{3 - \sqrt{21}}{6} & 0 \\ 0 & - \frac{21 - 5 \sqrt{21}}{3} & \frac{3 - \sqrt{21}}{3} & 0 \\ 3 - 3 \cdot 1 & - 11 - 3 \frac{3 - \sqrt{21}}{3} & 4 + \sqrt{21} - 3 (- \frac{3 - \sqrt{21}}{6}) & 0 - 3 \cdot 0 \end{array}]$

Etapa 5.3.2.3.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{3 - \sqrt{21}}{3} & - \frac{3 - \sqrt{21}}{6} & 0 \\ 0 & - \frac{21 - 5 \sqrt{21}}{3} & \frac{3 - \sqrt{21}}{3} & 0 \\ 0 & - 14 + \sqrt{21} & \frac{11 + \sqrt{21}}{2} & 0 \end{array}]$

Etapa 5.3.2.4

Multiply each element of $R_{2}$ by $- \frac{3}{21 - 5 \sqrt{21}}$ to make the entry at $2, 2$ a $1$ .

Toque para ver mais passagens...

Etapa 5.3.2.4.1

Multiply each element of $R_{2}$ by $- \frac{3}{21 - 5 \sqrt{21}}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{3 - \sqrt{21}}{3} & - \frac{3 - \sqrt{21}}{6} & 0 \\ - \frac{3}{21 - 5 \sqrt{21}} \cdot 0 & - \frac{3}{21 - 5 \sqrt{21}} (- \frac{21 - 5 \sqrt{21}}{3}) & - \frac{3}{21 - 5 \sqrt{21}} \cdot \frac{3 - \sqrt{21}}{3} & - \frac{3}{21 - 5 \sqrt{21}} \cdot 0 \\ 0 & - 14 + \sqrt{21} & \frac{11 + \sqrt{21}}{2} & 0 \end{array}]$

Etapa 5.3.2.4.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3 - \sqrt{21}}{3} & - \frac{3 - \sqrt{21}}{6} & 0 \\ 0 & 1 & - \frac{7 + \sqrt{21}}{14} & 0 \\ 0 & - 14 + \sqrt{21} & \frac{11 + \sqrt{21}}{2} & 0 \end{array}]$

Etapa 5.3.2.5

Perform the row operation $R_{3} = R_{3} - (- 14 + \sqrt{21}) R_{2}$ to make the entry at $3, 2$ a $0$ .

Toque para ver mais passagens...

Etapa 5.3.2.5.1

Perform the row operation $R_{3} = R_{3} - (- 14 + \sqrt{21}) R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{3 - \sqrt{21}}{3} & - \frac{3 - \sqrt{21}}{6} & 0 \\ 0 & 1 & - \frac{7 + \sqrt{21}}{14} & 0 \\ 0 - (- 14 + \sqrt{21}) \cdot 0 & - 14 + \sqrt{21} - (- 14 + \sqrt{21}) \cdot 1 & \frac{11 + \sqrt{21}}{2} - (- 14 + \sqrt{21}) (- \frac{7 + \sqrt{21}}{14}) & 0 - (- 14 + \sqrt{21}) \cdot 0 \end{array}]$

Etapa 5.3.2.5.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{3 - \sqrt{21}}{3} & - \frac{3 - \sqrt{21}}{6} & 0 \\ 0 & 1 & - \frac{7 + \sqrt{21}}{14} & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 5.3.2.6

Perform the row operation $R_{1} = R_{1} - \frac{3 - \sqrt{21}}{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

Toque para ver mais passagens...

Etapa 5.3.2.6.1

Perform the row operation $R_{1} = R_{1} - \frac{3 - \sqrt{21}}{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{3 - \sqrt{21}}{3} \cdot 0 & \frac{3 - \sqrt{21}}{3} - \frac{3 - \sqrt{21}}{3} \cdot 1 & - \frac{3 - \sqrt{21}}{6} - \frac{3 - \sqrt{21}}{3} (- \frac{7 + \sqrt{21}}{14}) & 0 - \frac{3 - \sqrt{21}}{3} \cdot 0 \\ 0 & 1 & - \frac{7 + \sqrt{21}}{14} & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 5.3.2.6.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - \frac{7 - \sqrt{21}}{14} & 0 \\ 0 & 1 & - \frac{7 + \sqrt{21}}{14} & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 5.3.3

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

$x - \frac{7 - \sqrt{21}}{14} z = 0$

$y - \frac{7 + \sqrt{21}}{14} z = 0$

$0 = 0$

Etapa 5.3.4

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

$[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} \frac{z}{2} - \frac{z \sqrt{21}}{14} \\ \frac{z}{2} + \frac{z \sqrt{21}}{14} \\ z \end{array}]$

Etapa 5.3.5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x \\ y \\ z \end{array}] = z [\begin{array}{c} \frac{1}{2} - \frac{\sqrt{21}}{14} \\ \frac{1}{2} + \frac{\sqrt{21}}{14} \\ 1 \end{array}]$

Etapa 5.3.6

Write as a solution set.

${z [\begin{array}{c} \frac{1}{2} - \frac{\sqrt{21}}{14} \\ \frac{1}{2} + \frac{\sqrt{21}}{14} \\ 1 \end{array}] | z \in R}$

Etapa 5.3.7

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

${[\begin{array}{c} \frac{1}{2} - \frac{\sqrt{21}}{14} \\ \frac{1}{2} + \frac{\sqrt{21}}{14} \\ 1 \end{array}]}$

Etapa 6

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

${[\begin{array}{c} 14 \\ 4 \\ 1 \end{array}], [\begin{array}{c} \frac{1}{2} + \frac{\sqrt{21}}{14} \\ \frac{1}{2} - \frac{\sqrt{21}}{14} \\ 1 \end{array}], [\begin{array}{c} \frac{1}{2} - \frac{\sqrt{21}}{14} \\ \frac{1}{2} + \frac{\sqrt{21}}{14} \\ 1 \end{array}]}$