Álgebra lineal Ejemplos

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

Paso 1

Simplifica cada término.

Toca para ver más pasos...

Paso 1.1

Multiplica $a$ por cada elemento de la matriz.

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

Paso 1.2

Simplifica cada elemento de la matriz.

Toca para ver más pasos...

Paso 1.2.1

Multiplica $a$ por $1$ .

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

Paso 1.2.2

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

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

Paso 1.2.3

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

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

Paso 1.2.4

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

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

Paso 1.3

Multiplica $b$ por cada elemento de la matriz.

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

Paso 1.4

Simplifica cada elemento de la matriz.

Toca para ver más pasos...

Paso 1.4.1

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

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

Paso 1.4.2

Multiplica $b$ por $1$ .

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

Paso 1.4.3

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

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

Paso 1.5

Multiplica $c$ por cada elemento de la matriz.

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

Paso 1.6

Simplifica cada elemento de la matriz.

Toca para ver más pasos...

Paso 1.6.1

Multiplica $c$ por $1$ .

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

Paso 1.6.2

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

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

Paso 1.6.3

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

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

Paso 1.7

Multiplica $d$ por cada elemento de la matriz.

Paso 1.8

Simplifica cada elemento de la matriz.

Toca para ver más pasos...

Paso 1.8.1

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

Paso 1.8.2

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

Paso 1.8.3

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

Paso 2

Suma los elementos correspondientes.

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

Paso 3

Suma los elementos correspondientes.

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

Paso 4

Suma los elementos correspondientes.

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

Paso 5

Write as a linear system of equations.

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

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

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

Paso 6

Reduce el sistema.

$10 = a + 2 b + c + 3 d, - 15 = - a + b + 3 c - 4 d, 131 = 8 a + 7 b - 4 c + 36 d$