Álgebra lineal Ejemplos

$w - x + y + z = - 5$ , $4 w - 3 x - 4 y + 2 z = - 37$ , $3 w + x - 3 y - 3 z = - 11$ , $- w + 2 x + 2 y + 2 z = 6$

Paso 1

Escribe el sistema de ecuaciones en forma de matriz.

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 4 & - 3 & - 4 & 2 & - 37 \\ 3 & 1 & - 3 & - 3 & - 11 \\ - 1 & 2 & 2 & 2 & 6 \end{array}]$

Paso 2

Obtén la forma escalonada reducida por filas.

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Paso 2.1

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

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Paso 2.1.1

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

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 4 - 4 \cdot 1 & - 3 - 4 \cdot - 1 & - 4 - 4 \cdot 1 & 2 - 4 \cdot 1 & - 37 - 4 \cdot - 5 \\ 3 & 1 & - 3 & - 3 & - 11 \\ - 1 & 2 & 2 & 2 & 6 \end{array}]$

Paso 2.1.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 0 & 1 & - 8 & - 2 & - 17 \\ 3 & 1 & - 3 & - 3 & - 11 \\ - 1 & 2 & 2 & 2 & 6 \end{array}]$

Paso 2.2

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

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Paso 2.2.1

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

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 0 & 1 & - 8 & - 2 & - 17 \\ 3 - 3 \cdot 1 & 1 - 3 \cdot - 1 & - 3 - 3 \cdot 1 & - 3 - 3 \cdot 1 & - 11 - 3 \cdot - 5 \\ - 1 & 2 & 2 & 2 & 6 \end{array}]$

Paso 2.2.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 0 & 1 & - 8 & - 2 & - 17 \\ 0 & 4 & - 6 & - 6 & 4 \\ - 1 & 2 & 2 & 2 & 6 \end{array}]$

Paso 2.3

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

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Paso 2.3.1

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

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 0 & 1 & - 8 & - 2 & - 17 \\ 0 & 4 & - 6 & - 6 & 4 \\ - 1 + 1 \cdot 1 & 2 - 1 & 2 + 1 \cdot 1 & 2 + 1 \cdot 1 & 6 - 5 \end{array}]$

Paso 2.3.2

Simplifica $R_{4}$ .

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 0 & 1 & - 8 & - 2 & - 17 \\ 0 & 4 & - 6 & - 6 & 4 \\ 0 & 1 & 3 & 3 & 1 \end{array}]$

Paso 2.4

Perform the row operation $R_{3} = R_{3} - 4 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Paso 2.4.1

Perform the row operation $R_{3} = R_{3} - 4 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 0 & 1 & - 8 & - 2 & - 17 \\ 0 - 4 \cdot 0 & 4 - 4 \cdot 1 & - 6 - 4 \cdot - 8 & - 6 - 4 \cdot - 2 & 4 - 4 \cdot - 17 \\ 0 & 1 & 3 & 3 & 1 \end{array}]$

Paso 2.4.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 0 & 1 & - 8 & - 2 & - 17 \\ 0 & 0 & 26 & 2 & 72 \\ 0 & 1 & 3 & 3 & 1 \end{array}]$

Paso 2.5

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

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Paso 2.5.1

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

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 0 & 1 & - 8 & - 2 & - 17 \\ 0 & 0 & 26 & 2 & 72 \\ 0 - 0 & 1 - 1 & 3 + 8 & 3 + 2 & 1 + 17 \end{array}]$

Paso 2.5.2

Simplifica $R_{4}$ .

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 0 & 1 & - 8 & - 2 & - 17 \\ 0 & 0 & 26 & 2 & 72 \\ 0 & 0 & 11 & 5 & 18 \end{array}]$

Paso 2.6

Multiply each element of $R_{3}$ by $\frac{1}{26}$ to make the entry at $3, 3$ a $1$ .

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Paso 2.6.1

Multiply each element of $R_{3}$ by $\frac{1}{26}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 0 & 1 & - 8 & - 2 & - 17 \\ \frac{0}{26} & \frac{0}{26} & \frac{26}{26} & \frac{2}{26} & \frac{72}{26} \\ 0 & 0 & 11 & 5 & 18 \end{array}]$

Paso 2.6.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 0 & 1 & - 8 & - 2 & - 17 \\ 0 & 0 & 1 & \frac{1}{13} & \frac{36}{13} \\ 0 & 0 & 11 & 5 & 18 \end{array}]$

Paso 2.7

Perform the row operation $R_{4} = R_{4} - 11 R_{3}$ to make the entry at $4, 3$ a $0$ .

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Paso 2.7.1

Perform the row operation $R_{4} = R_{4} - 11 R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 0 & 1 & - 8 & - 2 & - 17 \\ 0 & 0 & 1 & \frac{1}{13} & \frac{36}{13} \\ 0 - 11 \cdot 0 & 0 - 11 \cdot 0 & 11 - 11 \cdot 1 & 5 - 11 (\frac{1}{13}) & 18 - 11 (\frac{36}{13}) \end{array}]$

Paso 2.7.2

Simplifica $R_{4}$ .

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 0 & 1 & - 8 & - 2 & - 17 \\ 0 & 0 & 1 & \frac{1}{13} & \frac{36}{13} \\ 0 & 0 & 0 & \frac{54}{13} & - \frac{162}{13} \end{array}]$

Paso 2.8

Multiply each element of $R_{4}$ by $\frac{13}{54}$ to make the entry at $4, 4$ a $1$ .

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Paso 2.8.1

Multiply each element of $R_{4}$ by $\frac{13}{54}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 0 & 1 & - 8 & - 2 & - 17 \\ 0 & 0 & 1 & \frac{1}{13} & \frac{36}{13} \\ \frac{13}{54} \cdot 0 & \frac{13}{54} \cdot 0 & \frac{13}{54} \cdot 0 & \frac{13}{54} \cdot \frac{54}{13} & \frac{13}{54} (- \frac{162}{13}) \end{array}]$

Paso 2.8.2

Simplifica $R_{4}$ .

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 0 & 1 & - 8 & - 2 & - 17 \\ 0 & 0 & 1 & \frac{1}{13} & \frac{36}{13} \\ 0 & 0 & 0 & 1 & - 3 \end{array}]$

Paso 2.9

Perform the row operation $R_{3} = R_{3} - \frac{1}{13} R_{4}$ to make the entry at $3, 4$ a $0$ .

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Paso 2.9.1

Perform the row operation $R_{3} = R_{3} - \frac{1}{13} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 0 & 1 & - 8 & - 2 & - 17 \\ 0 - \frac{1}{13} \cdot 0 & 0 - \frac{1}{13} \cdot 0 & 1 - \frac{1}{13} \cdot 0 & \frac{1}{13} - \frac{1}{13} \cdot 1 & \frac{36}{13} - \frac{1}{13} \cdot - 3 \\ 0 & 0 & 0 & 1 & - 3 \end{array}]$

Paso 2.9.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 0 & 1 & - 8 & - 2 & - 17 \\ 0 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 1 & - 3 \end{array}]$

Paso 2.10

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

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Paso 2.10.1

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

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 0 + 2 \cdot 0 & 1 + 2 \cdot 0 & - 8 + 2 \cdot 0 & - 2 + 2 \cdot 1 & - 17 + 2 \cdot - 3 \\ 0 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 1 & - 3 \end{array}]$

Paso 2.10.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 1 & 1 & - 5 \\ 0 & 1 & - 8 & 0 & - 23 \\ 0 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 1 & - 3 \end{array}]$

Paso 2.11

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

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Paso 2.11.1

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

$[\begin{array}{c} 1 - 0 & - 1 - 0 & 1 - 0 & 1 - 1 & - 5 + 3 \\ 0 & 1 & - 8 & 0 & - 23 \\ 0 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 1 & - 3 \end{array}]$

Paso 2.11.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & - 1 & 1 & 0 & - 2 \\ 0 & 1 & - 8 & 0 & - 23 \\ 0 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 1 & - 3 \end{array}]$

Paso 2.12

Perform the row operation $R_{2} = R_{2} + 8 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Paso 2.12.1

Perform the row operation $R_{2} = R_{2} + 8 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - 1 & 1 & 0 & - 2 \\ 0 + 8 \cdot 0 & 1 + 8 \cdot 0 & - 8 + 8 \cdot 1 & 0 + 8 \cdot 0 & - 23 + 8 \cdot 3 \\ 0 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 1 & - 3 \end{array}]$

Paso 2.12.2

Simplifica $R_{2}$ .

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

Paso 2.13

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

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Paso 2.13.1

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

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

Paso 2.13.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & - 1 & 0 & 0 & - 5 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 1 & - 3 \end{array}]$

Paso 2.14

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

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Paso 2.14.1

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

$[\begin{array}{c} 1 + 0 & - 1 + 1 \cdot 1 & 0 + 0 & 0 + 0 & - 5 + 1 \cdot 1 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 1 & - 3 \end{array}]$

Paso 2.14.2

Simplifica $R_{1}$ .

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

Paso 3

Usa la matriz de resultados para declarar las soluciones finales en el sistema de ecuaciones.

$w = - 4$

$x = 1$

$y = 3$

$z = - 3$

Paso 4

La solución es el conjunto de pares ordenados que hacen que el sistema sea verdadero.

$(- 4, 1, 3, - 3)$

Paso 5

Descompone un vector de solución mediante la reorganización de cada ecuación representada en la forma reducida de fila de la matriz aumentada, a través de la resolución para la variable dependiente en cada fila, se obtiene la igualdad del vector.

$X = [\begin{array}{c} w \\ x \\ y \\ z \end{array}] = [\begin{array}{c} - 4 \\ 1 \\ 3 \\ - 3 \end{array}]$