Álgebra lineal Ejemplos

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

Paso 1

Obtén los valores propios.

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Paso 1.1

Establece la fórmula para obtener la ecuación característica

$p (λ)$ .

$p (λ) = determinante (A - λ I_{3})$

Paso 1.2

La matriz de identidades o matriz unidad de tamaño

$3$ es la matriz cuadrada

$3 \times 3$ con unos en la diagonal principal y ceros en los otros lugares.

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

Paso 1.3

Sustituye los valores conocidos en

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

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Paso 1.3.1

Sustituye

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

$A$ .

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

Paso 1.3.2

Sustituye

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

$I_{3}$ .

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

Paso 1.4

Simplifica.

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Paso 1.4.1

Simplifica cada término.

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Paso 1.4.1.1

Multiplica

$- λ$ por cada elemento de la matriz.

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

Paso 1.4.1.2

Simplifica cada elemento de la matriz.

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Paso 1.4.1.2.1

Multiplica

$- 1$ por

$1$ .

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

Paso 1.4.1.2.2

Multiplica

$- λ \cdot 0$ .

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Paso 1.4.1.2.2.1

Multiplica

$0$ por

$- 1$ .

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

Paso 1.4.1.2.2.2

Multiplica

$0$ por

$λ$ .

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

Paso 1.4.1.2.3

Multiplica

$- λ \cdot 0$ .

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Paso 1.4.1.2.3.1

Multiplica

$0$ por

$- 1$ .

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

Paso 1.4.1.2.3.2

Multiplica

$0$ por

$λ$ .

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

Paso 1.4.1.2.4

Multiplica

$- λ \cdot 0$ .

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Paso 1.4.1.2.4.1

Multiplica

$0$ por

$- 1$ .

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

Paso 1.4.1.2.4.2

Multiplica

$0$ por

$λ$ .

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

Paso 1.4.1.2.5

Multiplica

$- 1$ por

$1$ .

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

Paso 1.4.1.2.6

Multiplica

$- λ \cdot 0$ .

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Paso 1.4.1.2.6.1

Multiplica

$0$ por

$- 1$ .

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

Paso 1.4.1.2.6.2

Multiplica

$0$ por

$λ$ .

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

Paso 1.4.1.2.7

Multiplica

$- λ \cdot 0$ .

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Paso 1.4.1.2.7.1

Multiplica

$0$ por

$- 1$ .

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

Paso 1.4.1.2.7.2

Multiplica

$0$ por

$λ$ .

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

Paso 1.4.1.2.8

Multiplica

$- λ \cdot 0$ .

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Paso 1.4.1.2.8.1

Multiplica

$0$ por

$- 1$ .

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

Paso 1.4.1.2.8.2

Multiplica

$0$ por

$λ$ .

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

Paso 1.4.1.2.9

Multiplica

$- 1$ por

$1$ .

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

Paso 1.4.2

Suma los elementos correspondientes.

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

Paso 1.4.3

Simplify each element.

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Paso 1.4.3.1

Suma

$0.2$ y

$0$ .

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

Paso 1.4.3.2

Suma

$0.2$ y

$0$ .

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

Paso 1.4.3.3

Suma

$0$ y

$0$ .

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

Paso 1.4.3.4

Suma

$0.1$ y

$0$ .

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

Paso 1.4.3.5

Suma

$0.2$ y

$0$ .

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

Paso 1.4.3.6

Suma

$0.3$ y

$0$ .

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

Paso 1.5

Find the determinant.

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Paso 1.5.1

Choose the row or column with the most

$0$ elements. If there are no

$0$ elements choose any row or column. Multiply every element in column

$1$ by its cofactor and add.

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Paso 1.5.1.1

Consider the corresponding sign chart.

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

Paso 1.5.1.2

The cofactor is the minor with the sign changed if the indices match a

$-$ position on the sign chart.

Paso 1.5.1.3

The minor for

$a_{11}$ is the determinant with row

$1$ and column

$1$ deleted.

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

Paso 1.5.1.4

Multiply element

$a_{11}$ by its cofactor.

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

Paso 1.5.1.5

The minor for

$a_{21}$ is the determinant with row

$2$ and column

$1$ deleted.

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

Paso 1.5.1.6

Multiply element

$a_{21}$ by its cofactor.

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

Paso 1.5.1.7

The minor for

$a_{31}$ is the determinant with row

$3$ and column

$1$ deleted.

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

Paso 1.5.1.8

Multiply element

$a_{31}$ by its cofactor.

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

Paso 1.5.1.9

Add the terms together.

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

Paso 1.5.2

Multiplica

$0$ por

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

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

Paso 1.5.3

Evalúa

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

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Paso 1.5.3.1

El determinante de una matriz

$2 \times 2$ puede obtenerse usando la fórmula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

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

Paso 1.5.3.2

Simplifica el determinante.

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Paso 1.5.3.2.1

Simplifica cada término.

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Paso 1.5.3.2.1.1

Expande

$(0.5 - λ) (0.7 - λ)$ con el método PEIU (primero, exterior, interior, ultimo).

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Paso 1.5.3.2.1.1.1

Aplica la propiedad distributiva.

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

Paso 1.5.3.2.1.1.2

Aplica la propiedad distributiva.

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

Paso 1.5.3.2.1.1.3

Aplica la propiedad distributiva.

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

Paso 1.5.3.2.1.2

Simplifica y combina los términos similares.

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Paso 1.5.3.2.1.2.1

Simplifica cada término.

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Paso 1.5.3.2.1.2.1.1

Multiplica

$0.5$ por

$0.7$ .

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

Paso 1.5.3.2.1.2.1.2

Multiplica

$- 1$ por

$0.5$ .

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

Paso 1.5.3.2.1.2.1.3

Multiplica

$0.7$ por

$- 1$ .

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

Paso 1.5.3.2.1.2.1.4

Reescribe con la propiedad conmutativa de la multiplicación.

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

Paso 1.5.3.2.1.2.1.5

Multiplica

$λ$ por

$λ$ sumando los exponentes.

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Paso 1.5.3.2.1.2.1.5.1

Mueve

$λ$ .

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

Paso 1.5.3.2.1.2.1.5.2

Multiplica

$λ$ por

$λ$ .

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

Paso 1.5.3.2.1.2.1.6

Multiplica

$- 1$ por

$- 1$ .

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

Paso 1.5.3.2.1.2.1.7

Multiplica

$λ^{2}$ por

$1$ .

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

Paso 1.5.3.2.1.2.2

Resta

$0.7 λ$ de

$- 0.5 λ$ .

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

Paso 1.5.3.2.1.3

Multiplica

$- 0.3$ por

$0.1$ .

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

Paso 1.5.3.2.2

Resta

$0.03$ de

$0.35$ .

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

Paso 1.5.3.2.3

Reordena

$- 1.2 λ$ y

$λ^{2}$ .

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

Paso 1.5.4

Evalúa

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

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Paso 1.5.4.1

El determinante de una matriz

$2 \times 2$ puede obtenerse usando la fórmula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

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

Paso 1.5.4.2

Simplifica el determinante.

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Paso 1.5.4.2.1

Simplifica cada término.

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Paso 1.5.4.2.1.1

Multiplica

$0.2$ por

$0.1$ .

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

Paso 1.5.4.2.1.2

Aplica la propiedad distributiva.

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

Paso 1.5.4.2.1.3

Multiplica

$- 1$ por

$0.5$ .

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

Paso 1.5.4.2.1.4

Multiplica

$- - λ$ .

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Paso 1.5.4.2.1.4.1

Multiplica

$- 1$ por

$- 1$ .

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

Paso 1.5.4.2.1.4.2

Multiplica

$λ$ por

$1$ .

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

Paso 1.5.4.2.1.5

Aplica la propiedad distributiva.

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

Paso 1.5.4.2.1.6

Multiplica

$- 0.5$ por

$0.2$ .

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

Paso 1.5.4.2.1.7

Mueve

$0.2$ a la izquierda de

$λ$ .

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

Paso 1.5.4.2.2

Resta

$0.1$ de

$0.02$ .

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

Paso 1.5.4.2.3

Reordena

$- 0.08$ y

$0.2 λ$ .

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

Paso 1.5.5

Simplifica el determinante.

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Paso 1.5.5.1

Suma

$(0.8 - λ) (λ^{2} - 1.2 λ + 0.32)$ y

$0$ .

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

Paso 1.5.5.2

Simplifica cada término.

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Paso 1.5.5.2.1

Expande

$(0.8 - λ) (λ^{2} - 1.2 λ + 0.32)$ mediante la multiplicación de cada término de la primera expresión por cada término de la segunda expresión.

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

Paso 1.5.5.2.2

Simplifica cada término.

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Paso 1.5.5.2.2.1

Multiplica

$- 1.2$ por

$0.8$ .

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

Paso 1.5.5.2.2.2

Multiplica

$0.8$ por

$0.32$ .

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

Paso 1.5.5.2.2.3

Multiplica

$λ$ por

$λ^{2}$ sumando los exponentes.

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Paso 1.5.5.2.2.3.1

Mueve

$λ^{2}$ .

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

Paso 1.5.5.2.2.3.2

Multiplica

$λ^{2}$ por

$λ$ .

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Paso 1.5.5.2.2.3.2.1

Eleva

$λ$ a la potencia de

$1$ .

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

Paso 1.5.5.2.2.3.2.2

Usa la regla de la potencia

$a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

Paso 1.5.5.2.2.3.3

Suma

$2$ y

$1$ .

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

Paso 1.5.5.2.2.4

Reescribe con la propiedad conmutativa de la multiplicación.

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

Paso 1.5.5.2.2.5

Multiplica

$λ$ por

$λ$ sumando los exponentes.

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Paso 1.5.5.2.2.5.1

Mueve

$λ$ .

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

Paso 1.5.5.2.2.5.2

Multiplica

$λ$ por

$λ$ .

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

Paso 1.5.5.2.2.6

Multiplica

$- 1$ por

$- 1.2$ .

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

Paso 1.5.5.2.2.7

Multiplica

$0.32$ por

$- 1$ .

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

Paso 1.5.5.2.3

Suma

$0.8 λ^{2}$ y

$1.2 λ^{2}$ .

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

Paso 1.5.5.2.4

Resta

$0.32 λ$ de

$- 0.96 λ$ .

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

Paso 1.5.5.2.5

Aplica la propiedad distributiva.

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

Paso 1.5.5.2.6

Multiplica

$0.2$ por

$0.2$ .

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

Paso 1.5.5.2.7

Multiplica

$0.2$ por

$- 0.08$ .

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

Paso 1.5.5.3

Suma

$- 1.28 λ$ y

$0.04 λ$ .

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

Paso 1.5.5.4

Resta

$0.016$ de

$0.256$ .

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

Paso 1.5.5.5

Mueve

$- 1.24 λ$ .

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

Paso 1.5.5.6

Reordena

$2 λ^{2}$ y

$- λ^{3}$ .

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

Paso 1.6

Establece el polinomio característico igual a

$0$ para obtener los valores propios

$λ$ .

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

Paso 1.7

Resuelve

$λ$

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Paso 1.7.1

Grafica cada lado de la ecuación. La solución es el valor x del punto de intersección.

$λ = \frac{2}{5}, \frac{3}{5}, 1$

Paso 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.

$ε_{A} = N (A - λ I_{3})$

Paso 3

Find the eigenvector using the eigenvalue

$λ = \frac{2}{5}$ .

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Paso 3.1

Sustituye los valores conocidos en la fórmula.

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

Paso 3.2

Simplifica.

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Paso 3.2.1

Simplifica cada término.

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Paso 3.2.1.1

Multiplica

$- \frac{2}{5}$ por cada elemento de la matriz.

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

Paso 3.2.1.2

Simplifica cada elemento de la matriz.

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Paso 3.2.1.2.1

Multiplica

$- 1$ por

$1$ .

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

Paso 3.2.1.2.2

Multiplica

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

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Paso 3.2.1.2.2.1

Multiplica

$0$ por

$- 1$ .

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

Paso 3.2.1.2.2.2

Multiplica

$0$ por

$\frac{2}{5}$ .

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

Paso 3.2.1.2.3

Multiplica

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

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Paso 3.2.1.2.3.1

Multiplica

$0$ por

$- 1$ .

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

Paso 3.2.1.2.3.2

Multiplica

$0$ por

$\frac{2}{5}$ .

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

Paso 3.2.1.2.4

Multiplica

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

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Paso 3.2.1.2.4.1

Multiplica

$0$ por

$- 1$ .

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

Paso 3.2.1.2.4.2

Multiplica

$0$ por

$\frac{2}{5}$ .

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

Paso 3.2.1.2.5

Multiplica

$- 1$ por

$1$ .

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

Paso 3.2.1.2.6

Multiplica

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

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Paso 3.2.1.2.6.1

Multiplica

$0$ por

$- 1$ .

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

Paso 3.2.1.2.6.2

Multiplica

$0$ por

$\frac{2}{5}$ .

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

Paso 3.2.1.2.7

Multiplica

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

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Paso 3.2.1.2.7.1

Multiplica

$0$ por

$- 1$ .

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

Paso 3.2.1.2.7.2

Multiplica

$0$ por

$\frac{2}{5}$ .

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

Paso 3.2.1.2.8

Multiplica

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

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Paso 3.2.1.2.8.1

Multiplica

$0$ por

$- 1$ .

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

Paso 3.2.1.2.8.2

Multiplica

$0$ por

$\frac{2}{5}$ .

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

Paso 3.2.1.2.9

Multiplica

$- 1$ por

$1$ .

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

Paso 3.2.2

Suma los elementos correspondientes.

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

Paso 3.2.3

Simplify each element.

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Paso 3.2.3.1

Para escribir

$0.8$ como una fracción con un denominador común, multiplica por

$\frac{5}{5}$ .

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

Paso 3.2.3.2

Combina

$0.8$ y

$\frac{5}{5}$ .

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

Paso 3.2.3.3

Combina los numeradores sobre el denominador común.

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

Paso 3.2.3.4

Simplifica el numerador.

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Paso 3.2.3.4.1

Multiplica

$0.8$ por

$5$ .

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

Paso 3.2.3.4.2

Resta

$2$ de

$4$ .

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

Paso 3.2.3.5

Cancela el factor común de

$2$ y

$5$ .

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Paso 3.2.3.5.1

Reescribe

$2$ como

$1 (2)$ .

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

Paso 3.2.3.5.2

Cancela los factores comunes.

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Paso 3.2.3.5.2.1

Reescribe

$5$ como

$1 (5)$ .

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

Paso 3.2.3.5.2.2

Cancela el factor común.

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

Paso 3.2.3.5.2.3

Reescribe la expresión.

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

Paso 3.2.3.6

Suma

$0.2$ y

$0$ .

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

Paso 3.2.3.7

Suma

$0.2$ y

$0$ .

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

Paso 3.2.3.8

Suma

$0$ y

$0$ .

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

Paso 3.2.3.9

Para escribir

$0.5$ como una fracción con un denominador común, multiplica por

$\frac{5}{5}$ .

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

Paso 3.2.3.10

Combina

$0.5$ y

$\frac{5}{5}$ .

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

Paso 3.2.3.11

Combina los numeradores sobre el denominador común.

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

Paso 3.2.3.12

Simplifica el numerador.

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Paso 3.2.3.12.1

Multiplica

$0.5$ por

$5$ .

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

Paso 3.2.3.12.2

Resta

$2$ de

$2.5$ .

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

Paso 3.2.3.13

Divide

$0.5$ por

$5$ .

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

Paso 3.2.3.14

Suma

$0.1$ y

$0$ .

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

Paso 3.2.3.15

Suma

$0.2$ y

$0$ .

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

Paso 3.2.3.16

Suma

$0.3$ y

$0$ .

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

Paso 3.2.3.17

Para escribir

$0.7$ como una fracción con un denominador común, multiplica por

$\frac{5}{5}$ .

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

Paso 3.2.3.18

Combina

$0.7$ y

$\frac{5}{5}$ .

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

Paso 3.2.3.19

Combina los numeradores sobre el denominador común.

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

Paso 3.2.3.20

Simplifica el numerador.

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Paso 3.2.3.20.1

Multiplica

$0.7$ por

$5$ .

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

Paso 3.2.3.20.2

Resta

$2$ de

$3.5$ .

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

Paso 3.2.3.21

Divide

$1.5$ por

$5$ .

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

Paso 3.3

Find the null space when

$λ = \frac{2}{5}$ .

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Paso 3.3.1

Write as an augmented matrix for

$A x = 0$ .

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

Paso 3.3.2

Obtén la forma escalonada reducida por filas.

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Paso 3.3.2.1

Multiply each element of

$R_{1}$ by

$\frac{5}{2}$ to make the entry at

$1, 1$ a

$1$ .

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Paso 3.3.2.1.1

Multiply each element of

$R_{1}$ by

$\frac{5}{2}$ to make the entry at

$1, 1$ a

$1$ .

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

Paso 3.3.2.1.2

Simplifica

$R_{1}$ .

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

Paso 3.3.2.2

Perform the row operation

$R_{3} = R_{3} - 0.2 R_{1}$ to make the entry at

$3, 1$ a

$0$ .

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Paso 3.3.2.2.1

Perform the row operation

$R_{3} = R_{3} - 0.2 R_{1}$ to make the entry at

$3, 1$ a

$0$ .

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

Paso 3.3.2.2.2

Simplifica

$R_{3}$ .

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

Paso 3.3.2.3

Multiply each element of

$R_{2}$ by

$\frac{1}{0.1}$ to make the entry at

$2, 2$ a

$1$ .

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Paso 3.3.2.3.1

Multiply each element of

$R_{2}$ by

$\frac{1}{0.1}$ to make the entry at

$2, 2$ a

$1$ .

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

Paso 3.3.2.3.2

Simplifica

$R_{2}$ .

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

Paso 3.3.2.4

Perform the row operation

$R_{3} = R_{3} - \frac{1}{5} R_{2}$ to make the entry at

$3, 2$ a

$0$ .

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Paso 3.3.2.4.1

Perform the row operation

$R_{3} = R_{3} - \frac{1}{5} R_{2}$ to make the entry at

$3, 2$ a

$0$ .

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

Paso 3.3.2.4.2

Simplifica

$R_{3}$ .

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

Paso 3.3.2.5

Perform the row operation

$R_{1} = R_{1} - \frac{1}{2} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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Paso 3.3.2.5.1

Perform the row operation

$R_{1} = R_{1} - \frac{1}{2} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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

Paso 3.3.2.5.2

Simplifica

$R_{1}$ .

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

Paso 3.3.3

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

$x = 0$

$y + z = 0$

$0 = 0$

Paso 3.3.4

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

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

Paso 3.3.5

Write the solution as a linear combination of vectors.

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

Paso 3.3.6

Write as a solution set.

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

Paso 3.3.7

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

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

Paso 4

Find the eigenvector using the eigenvalue

$λ = \frac{3}{5}$ .

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Paso 4.1

Sustituye los valores conocidos en la fórmula.

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

Paso 4.2

Simplifica.

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Paso 4.2.1

Simplifica cada término.

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Paso 4.2.1.1

Multiplica

$- \frac{3}{5}$ por cada elemento de la matriz.

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

Paso 4.2.1.2

Simplifica cada elemento de la matriz.

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Paso 4.2.1.2.1

Multiplica

$- 1$ por

$1$ .

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

Paso 4.2.1.2.2

Multiplica

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

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Paso 4.2.1.2.2.1

Multiplica

$0$ por

$- 1$ .

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

Paso 4.2.1.2.2.2

Multiplica

$0$ por

$\frac{3}{5}$ .

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

Paso 4.2.1.2.3

Multiplica

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

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Paso 4.2.1.2.3.1

Multiplica

$0$ por

$- 1$ .

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

Paso 4.2.1.2.3.2

Multiplica

$0$ por

$\frac{3}{5}$ .

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

Paso 4.2.1.2.4

Multiplica

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

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Paso 4.2.1.2.4.1

Multiplica

$0$ por

$- 1$ .

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

Paso 4.2.1.2.4.2

Multiplica

$0$ por

$\frac{3}{5}$ .

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

Paso 4.2.1.2.5

Multiplica

$- 1$ por

$1$ .

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

Paso 4.2.1.2.6

Multiplica

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

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Paso 4.2.1.2.6.1

Multiplica

$0$ por

$- 1$ .

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

Paso 4.2.1.2.6.2

Multiplica

$0$ por

$\frac{3}{5}$ .

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

Paso 4.2.1.2.7

Multiplica

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

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Paso 4.2.1.2.7.1

Multiplica

$0$ por

$- 1$ .

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

Paso 4.2.1.2.7.2

Multiplica

$0$ por

$\frac{3}{5}$ .

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

Paso 4.2.1.2.8

Multiplica

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

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Paso 4.2.1.2.8.1

Multiplica

$0$ por

$- 1$ .

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

Paso 4.2.1.2.8.2

Multiplica

$0$ por

$\frac{3}{5}$ .

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

Paso 4.2.1.2.9

Multiplica

$- 1$ por

$1$ .

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

Paso 4.2.2

Suma los elementos correspondientes.

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

Paso 4.2.3

Simplify each element.

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Paso 4.2.3.1

Para escribir

$0.8$ como una fracción con un denominador común, multiplica por

$\frac{5}{5}$ .

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

Paso 4.2.3.2

Combina

$0.8$ y

$\frac{5}{5}$ .

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

Paso 4.2.3.3

Combina los numeradores sobre el denominador común.

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

Paso 4.2.3.4

Simplifica el numerador.

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Paso 4.2.3.4.1

Multiplica

$0.8$ por

$5$ .

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

Paso 4.2.3.4.2

Resta

$3$ de

$4$ .

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

Paso 4.2.3.5

Cancela el factor común de

$1$ y

$5$ .

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Paso 4.2.3.5.1

Reescribe

$1$ como

$1 (1)$ .

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

Paso 4.2.3.5.2

Cancela los factores comunes.

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Paso 4.2.3.5.2.1

Reescribe

$5$ como

$1 (5)$ .

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

Paso 4.2.3.5.2.2

Cancela el factor común.

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

Paso 4.2.3.5.2.3

Reescribe la expresión.

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

Paso 4.2.3.6

Suma

$0.2$ y

$0$ .

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

Paso 4.2.3.7

Suma

$0.2$ y

$0$ .

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

Paso 4.2.3.8

Suma

$0$ y

$0$ .

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

Paso 4.2.3.9

Para escribir

$0.5$ como una fracción con un denominador común, multiplica por

$\frac{5}{5}$ .

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

Paso 4.2.3.10

Combina

$0.5$ y

$\frac{5}{5}$ .

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

Paso 4.2.3.11

Combina los numeradores sobre el denominador común.

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

Paso 4.2.3.12

Simplifica el numerador.

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Paso 4.2.3.12.1

Multiplica

$0.5$ por

$5$ .

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

Paso 4.2.3.12.2

Resta

$3$ de

$2.5$ .

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

Paso 4.2.3.13

Divide

$- 0.5$ por

$5$ .

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

Paso 4.2.3.14

Suma

$0.1$ y

$0$ .

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

Paso 4.2.3.15

Suma

$0.2$ y

$0$ .

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

Paso 4.2.3.16

Suma

$0.3$ y

$0$ .

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

Paso 4.2.3.17

Para escribir

$0.7$ como una fracción con un denominador común, multiplica por

$\frac{5}{5}$ .

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

Paso 4.2.3.18

Combina

$0.7$ y

$\frac{5}{5}$ .

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

Paso 4.2.3.19

Combina los numeradores sobre el denominador común.

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

Paso 4.2.3.20

Simplifica el numerador.

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Paso 4.2.3.20.1

Multiplica

$0.7$ por

$5$ .

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

Paso 4.2.3.20.2

Resta

$3$ de

$3.5$ .

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

Paso 4.2.3.21

Divide

$0.5$ por

$5$ .

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

Paso 4.3

Find the null space when

$λ = \frac{3}{5}$ .

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Paso 4.3.1

Write as an augmented matrix for

$A x = 0$ .

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

Paso 4.3.2

Obtén la forma escalonada reducida por filas.

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Paso 4.3.2.1

Multiply each element of

$R_{1}$ by

$5$ to make the entry at

$1, 1$ a

$1$ .

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Paso 4.3.2.1.1

Multiply each element of

$R_{1}$ by

$5$ to make the entry at

$1, 1$ a

$1$ .

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

Paso 4.3.2.1.2

Simplifica

$R_{1}$ .

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

Paso 4.3.2.2

Perform the row operation

$R_{3} = R_{3} - 0.2 R_{1}$ to make the entry at

$3, 1$ a

$0$ .

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Paso 4.3.2.2.1

Perform the row operation

$R_{3} = R_{3} - 0.2 R_{1}$ to make the entry at

$3, 1$ a

$0$ .

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

Paso 4.3.2.2.2

Simplifica

$R_{3}$ .

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

Paso 4.3.2.3

Multiply each element of

$R_{2}$ by

$\frac{1}{- 0.1}$ to make the entry at

$2, 2$ a

$1$ .

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Paso 4.3.2.3.1

Multiply each element of

$R_{2}$ by

$\frac{1}{- 0.1}$ to make the entry at

$2, 2$ a

$1$ .

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

Paso 4.3.2.3.2

Simplifica

$R_{2}$ .

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

Paso 4.3.2.4

Perform the row operation

$R_{3} = R_{3} - 0.1 R_{2}$ to make the entry at

$3, 2$ a

$0$ .

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Paso 4.3.2.4.1

Perform the row operation

$R_{3} = R_{3} - 0.1 R_{2}$ to make the entry at

$3, 2$ a

$0$ .

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

Paso 4.3.2.4.2

Simplifica

$R_{3}$ .

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

Paso 4.3.2.5

Perform the row operation

$R_{1} = R_{1} - R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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Paso 4.3.2.5.1

Perform the row operation

$R_{1} = R_{1} - R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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

Paso 4.3.2.5.2

Simplifica

$R_{1}$ .

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

Paso 4.3.3

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

$x + 2 z = 0$

$y - z = 0$

$0 = 0$

Paso 4.3.4

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

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

Paso 4.3.5

Write the solution as a linear combination of vectors.

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

Paso 4.3.6

Write as a solution set.

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

Paso 4.3.7

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

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

Paso 5

Find the eigenvector using the eigenvalue

$λ = 1$ .

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Paso 5.1

Sustituye los valores conocidos en la fórmula.

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

Paso 5.2

Simplifica.

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Paso 5.2.1

Resta los elementos correspondientes.

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

Paso 5.2.2

Simplify each element.

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Paso 5.2.2.1

Resta

$1$ de

$0.8$ .

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

Paso 5.2.2.2

Resta

$0$ de

$0.2$ .

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

Paso 5.2.2.3

Resta

$0$ de

$0.2$ .

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

Paso 5.2.2.4

Resta

$0$ de

$0$ .

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

Paso 5.2.2.5

Resta

$1$ de

$0.5$ .

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

Paso 5.2.2.6

Resta

$0$ de

$0.1$ .

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

Paso 5.2.2.7

Resta

$0$ de

$0.2$ .

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

Paso 5.2.2.8

Resta

$0$ de

$0.3$ .

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

Paso 5.2.2.9

Resta

$1$ de

$0.7$ .

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

Paso 5.3

Find the null space when

$λ = 1$ .

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Paso 5.3.1

Write as an augmented matrix for

$A x = 0$ .

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

Paso 5.3.2

Obtén la forma escalonada reducida por filas.

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Paso 5.3.2.1

Multiply each element of

$R_{1}$ by

$\frac{1}{- 0.2}$ to make the entry at

$1, 1$ a

$1$ .

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Paso 5.3.2.1.1

Multiply each element of

$R_{1}$ by

$\frac{1}{- 0.2}$ to make the entry at

$1, 1$ a

$1$ .

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

Paso 5.3.2.1.2

Simplifica

$R_{1}$ .

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

Paso 5.3.2.2

Perform the row operation

$R_{3} = R_{3} - 0.2 R_{1}$ to make the entry at

$3, 1$ a

$0$ .

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Paso 5.3.2.2.1

Perform the row operation

$R_{3} = R_{3} - 0.2 R_{1}$ to make the entry at

$3, 1$ a

$0$ .

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

Paso 5.3.2.2.2

Simplifica

$R_{3}$ .

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

Paso 5.3.2.3

Multiply each element of

$R_{2}$ by

$\frac{1}{- 0.5}$ to make the entry at

$2, 2$ a

$1$ .

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Paso 5.3.2.3.1

Multiply each element of

$R_{2}$ by

$\frac{1}{- 0.5}$ to make the entry at

$2, 2$ a

$1$ .

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

Paso 5.3.2.3.2

Simplifica

$R_{2}$ .

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

Paso 5.3.2.4

Perform the row operation

$R_{3} = R_{3} - 0.5 R_{2}$ to make the entry at

$3, 2$ a

$0$ .

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Paso 5.3.2.4.1

Perform the row operation

$R_{3} = R_{3} - 0.5 R_{2}$ to make the entry at

$3, 2$ a

$0$ .

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

Paso 5.3.2.4.2

Simplifica

$R_{3}$ .

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

Paso 5.3.2.5

Perform the row operation

$R_{1} = R_{1} + R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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Paso 5.3.2.5.1

Perform the row operation

$R_{1} = R_{1} + R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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

Paso 5.3.2.5.2

Simplifica

$R_{1}$ .

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

Paso 5.3.3

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

$x - 1.2 z = 0$

$y - 0.2 z = 0$

$0 = 0$

Paso 5.3.4

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

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

Paso 5.3.5

Write the solution as a linear combination of vectors.

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

Paso 5.3.6

Write as a solution set.

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

Paso 5.3.7

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

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

Paso 6

The eigenspace of

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

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