Álgebra lineal Ejemplos

$a [\begin{array}{c} 1 \\ - 1 \\ 8 \end{array}] + b [\begin{array}{c} 2 \\ 1 \\ 7 \end{array}] + d [\begin{array}{c} 1 \\ 3 \\ - 4 \end{array}] + g [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 1

Simplifica.

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Paso 1.1

Simplifica cada término.

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Paso 1.1.1

Multiplica $a$ por cada elemento de la matriz.

$[\begin{array}{c} a \cdot 1 \\ a \cdot - 1 \\ a \cdot 8 \end{array}] + b [\begin{array}{c} 2 \\ 1 \\ 7 \end{array}] + d [\begin{array}{c} 1 \\ 3 \\ - 4 \end{array}] + g [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 1.1.2

Simplifica cada elemento de la matriz.

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Paso 1.1.2.1

Multiplica $a$ por $1$ .

$[\begin{array}{c} a \\ a \cdot - 1 \\ a \cdot 8 \end{array}] + b [\begin{array}{c} 2 \\ 1 \\ 7 \end{array}] + d [\begin{array}{c} 1 \\ 3 \\ - 4 \end{array}] + g [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 1.1.2.2

Mueve $- 1$ a la izquierda de $a$ .

$[\begin{array}{c} a \\ - 1 \cdot a \\ a \cdot 8 \end{array}] + b [\begin{array}{c} 2 \\ 1 \\ 7 \end{array}] + d [\begin{array}{c} 1 \\ 3 \\ - 4 \end{array}] + g [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 1.1.2.3

Reescribe $- 1 a$ como $- a$ .

$[\begin{array}{c} a \\ - a \\ a \cdot 8 \end{array}] + b [\begin{array}{c} 2 \\ 1 \\ 7 \end{array}] + d [\begin{array}{c} 1 \\ 3 \\ - 4 \end{array}] + g [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 1.1.2.4

Mueve $8$ a la izquierda de $a$ .

$[\begin{array}{c} a \\ - a \\ 8 a \end{array}] + b [\begin{array}{c} 2 \\ 1 \\ 7 \end{array}] + d [\begin{array}{c} 1 \\ 3 \\ - 4 \end{array}] + g [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 1.1.3

Multiplica $b$ por cada elemento de la matriz.

$[\begin{array}{c} a \\ - a \\ 8 a \end{array}] + [\begin{array}{c} b \cdot 2 \\ b \cdot 1 \\ b \cdot 7 \end{array}] + d [\begin{array}{c} 1 \\ 3 \\ - 4 \end{array}] + g [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 1.1.4

Simplifica cada elemento de la matriz.

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Paso 1.1.4.1

Mueve $2$ a la izquierda de $b$ .

$[\begin{array}{c} a \\ - a \\ 8 a \end{array}] + [\begin{array}{c} 2 b \\ b \cdot 1 \\ b \cdot 7 \end{array}] + d [\begin{array}{c} 1 \\ 3 \\ - 4 \end{array}] + g [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 1.1.4.2

Multiplica $b$ por $1$ .

$[\begin{array}{c} a \\ - a \\ 8 a \end{array}] + [\begin{array}{c} 2 b \\ b \\ b \cdot 7 \end{array}] + d [\begin{array}{c} 1 \\ 3 \\ - 4 \end{array}] + g [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 1.1.4.3

Mueve $7$ a la izquierda de $b$ .

$[\begin{array}{c} a \\ - a \\ 8 a \end{array}] + [\begin{array}{c} 2 b \\ b \\ 7 b \end{array}] + d [\begin{array}{c} 1 \\ 3 \\ - 4 \end{array}] + g [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 1.1.5

Multiplica $d$ por cada elemento de la matriz.

$[\begin{array}{c} a \\ - a \\ 8 a \end{array}] + [\begin{array}{c} 2 b \\ b \\ 7 b \end{array}] + [\begin{array}{c} d \cdot 1 \\ d \cdot 3 \\ d \cdot - 4 \end{array}] + g [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 1.1.6

Simplifica cada elemento de la matriz.

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Paso 1.1.6.1

Multiplica $d$ por $1$ .

$[\begin{array}{c} a \\ - a \\ 8 a \end{array}] + [\begin{array}{c} 2 b \\ b \\ 7 b \end{array}] + [\begin{array}{c} d \\ d \cdot 3 \\ d \cdot - 4 \end{array}] + g [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 1.1.6.2

Mueve $3$ a la izquierda de $d$ .

$[\begin{array}{c} a \\ - a \\ 8 a \end{array}] + [\begin{array}{c} 2 b \\ b \\ 7 b \end{array}] + [\begin{array}{c} d \\ 3 d \\ d \cdot - 4 \end{array}] + g [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 1.1.6.3

Mueve $- 4$ a la izquierda de $d$ .

$[\begin{array}{c} a \\ - a \\ 8 a \end{array}] + [\begin{array}{c} 2 b \\ b \\ 7 b \end{array}] + [\begin{array}{c} d \\ 3 d \\ - 4 d \end{array}] + g [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 1.1.7

Multiplica $g$ por cada elemento de la matriz.

$[\begin{array}{c} a \\ - a \\ 8 a \end{array}] + [\begin{array}{c} 2 b \\ b \\ 7 b \end{array}] + [\begin{array}{c} d \\ 3 d \\ - 4 d \end{array}] + [\begin{array}{c} g \cdot 3 \\ g \cdot - 4 \\ g \cdot 36 \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 1.1.8

Simplifica cada elemento de la matriz.

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Paso 1.1.8.1

Mueve $3$ a la izquierda de $g$ .

$[\begin{array}{c} a \\ - a \\ 8 a \end{array}] + [\begin{array}{c} 2 b \\ b \\ 7 b \end{array}] + [\begin{array}{c} d \\ 3 d \\ - 4 d \end{array}] + [\begin{array}{c} 3 g \\ g \cdot - 4 \\ g \cdot 36 \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 1.1.8.2

Mueve $- 4$ a la izquierda de $g$ .

$[\begin{array}{c} a \\ - a \\ 8 a \end{array}] + [\begin{array}{c} 2 b \\ b \\ 7 b \end{array}] + [\begin{array}{c} d \\ 3 d \\ - 4 d \end{array}] + [\begin{array}{c} 3 g \\ - 4 g \\ g \cdot 36 \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 1.1.8.3

Mueve $36$ a la izquierda de $g$ .

$[\begin{array}{c} a \\ - a \\ 8 a \end{array}] + [\begin{array}{c} 2 b \\ b \\ 7 b \end{array}] + [\begin{array}{c} d \\ 3 d \\ - 4 d \end{array}] + [\begin{array}{c} 3 g \\ - 4 g \\ 36 g \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 1.2

Suma los elementos correspondientes.

$[\begin{array}{c} a + 2 b \\ - a + b \\ 8 a + 7 b \end{array}] + [\begin{array}{c} d \\ 3 d \\ - 4 d \end{array}] + [\begin{array}{c} 3 g \\ - 4 g \\ 36 g \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 1.3

Suma los elementos correspondientes.

$[\begin{array}{c} a + 2 b + d \\ - a + b + 3 d \\ 8 a + 7 b - 4 d \end{array}] + [\begin{array}{c} 3 g \\ - 4 g \\ 36 g \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 1.4

Suma los elementos correspondientes.

$[\begin{array}{c} a + 2 b + d + 3 g \\ - a + b + 3 d - 4 g \\ 8 a + 7 b - 4 d + 36 g \end{array}] = [\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}]$

Paso 2

La ecuación de matriz puede escribirse como un conjunto de ecuaciones.

$a + 2 b + d + 3 g = 10$

$- a + b + 3 d - 4 g = - 15$

$8 a + 7 b - 4 d + 36 g = 131$