Álgebra lineal Ejemplos

$[\begin{array}{c} \cos (b) & \frac{i}{g} \cdot \sin (b) \\ \sin (b) \cdot (i g) & \cos (b) \end{array}] \cdot [\begin{array}{c} \cos (a) & \frac{i}{d} \cdot \sin (a) \\ \sin (a) \cdot (i d) & \cos (a) \end{array}]$

Paso 1

Combina $\frac{i}{g}$ y $\sin (b)$ .

$[\begin{array}{c} \cos (b) & \frac{i \sin (b)}{g} \\ \sin (b) \cdot (i g) & \cos (b) \end{array}] \cdot [\begin{array}{c} \cos (a) & \frac{i}{d} \cdot \sin (a) \\ \sin (a) \cdot (i d) & \cos (a) \end{array}]$

Paso 2

Reescribe con la propiedad conmutativa de la multiplicación.

$[\begin{array}{c} \cos (b) & \frac{i \sin (b)}{g} \\ i \sin (b) g & \cos (b) \end{array}] \cdot [\begin{array}{c} \cos (a) & \frac{i}{d} \cdot \sin (a) \\ \sin (a) \cdot (i d) & \cos (a) \end{array}]$

Paso 3

Combina $\frac{i}{d}$ y $\sin (a)$ .

$[\begin{array}{c} \cos (b) & \frac{i \sin (b)}{g} \\ i \sin (b) g & \cos (b) \end{array}] \cdot [\begin{array}{c} \cos (a) & \frac{i \sin (a)}{d} \\ \sin (a) \cdot (i d) & \cos (a) \end{array}]$

Paso 4

Reescribe con la propiedad conmutativa de la multiplicación.

$[\begin{array}{c} \cos (b) & \frac{i \sin (b)}{g} \\ i \sin (b) g & \cos (b) \end{array}] \cdot [\begin{array}{c} \cos (a) & \frac{i \sin (a)}{d} \\ i \sin (a) d & \cos (a) \end{array}]$

Paso 5

Multiplica $[\begin{array}{c} \cos (b) & \frac{i \sin (b)}{g} \\ i \sin (b) g & \cos (b) \end{array}] [\begin{array}{c} \cos (a) & \frac{i \sin (a)}{d} \\ i \sin (a) d & \cos (a) \end{array}]$ .

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Paso 5.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $2 \times 2$ and the second matrix is $2 \times 2$ .

Paso 5.2

Multiplica cada fila en la primera matriz por cada columna en la segunda matriz.

$[\begin{array}{c} \cos (b) \cos (a) + \frac{i \sin (b)}{g} (i \sin (a) d) & \cos (b) \frac{i \sin (a)}{d} + \frac{i \sin (b)}{g} \cos (a) \\ i \sin (b) g \cos (a) + \cos (b) (i \sin (a) d) & i \sin (b) g \frac{i \sin (a)}{d} + \cos (b) \cos (a) \end{array}]$

Paso 5.3

Simplifica cada elemento de la matriz mediante la multiplicación de todas las expresiones.

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Paso 5.3.1

Multiplica $i \sin (b) g \frac{i \sin (a)}{d}$ .

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Paso 5.3.1.1

Combina $i$ y $\frac{i \sin (a)}{d}$ .

$[\begin{array}{c} \cos (b) \cos (a) - \frac{d \sin (a) \sin (b)}{g} & \frac{i \cos (b) \sin (a)}{d} + \frac{i \sin (b) \cos (a)}{g} \\ i g \sin (b) \cos (a) + i d \cos (b) \sin (a) & \sin (b) g \frac{i (i \sin (a))}{d} + \cos (b) \cos (a) \end{array}]$

Paso 5.3.1.2

Eleva $i$ a la potencia de $1$ .

$[\begin{array}{c} \cos (b) \cos (a) - \frac{d \sin (a) \sin (b)}{g} & \frac{i \cos (b) \sin (a)}{d} + \frac{i \sin (b) \cos (a)}{g} \\ i g \sin (b) \cos (a) + i d \cos (b) \sin (a) & \sin (b) g \frac{i^{1} i \sin (a)}{d} + \cos (b) \cos (a) \end{array}]$

Paso 5.3.1.3

Eleva $i$ a la potencia de $1$ .

$[\begin{array}{c} \cos (b) \cos (a) - \frac{d \sin (a) \sin (b)}{g} & \frac{i \cos (b) \sin (a)}{d} + \frac{i \sin (b) \cos (a)}{g} \\ i g \sin (b) \cos (a) + i d \cos (b) \sin (a) & \sin (b) g \frac{i^{1} i^{1} \sin (a)}{d} + \cos (b) \cos (a) \end{array}]$

Paso 5.3.1.4

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

$[\begin{array}{c} \cos (b) \cos (a) - \frac{d \sin (a) \sin (b)}{g} & \frac{i \cos (b) \sin (a)}{d} + \frac{i \sin (b) \cos (a)}{g} \\ i g \sin (b) \cos (a) + i d \cos (b) \sin (a) & \sin (b) g \frac{i^{1 + 1} \sin (a)}{d} + \cos (b) \cos (a) \end{array}]$

Paso 5.3.1.5

Suma $1$ y $1$ .

$[\begin{array}{c} \cos (b) \cos (a) - \frac{d \sin (a) \sin (b)}{g} & \frac{i \cos (b) \sin (a)}{d} + \frac{i \sin (b) \cos (a)}{g} \\ i g \sin (b) \cos (a) + i d \cos (b) \sin (a) & \sin (b) g \frac{i^{2} \sin (a)}{d} + \cos (b) \cos (a) \end{array}]$

Paso 5.3.1.6

Combina $\sin (b)$ y $\frac{i^{2} \sin (a)}{d}$ .

$[\begin{array}{c} \cos (b) \cos (a) - \frac{d \sin (a) \sin (b)}{g} & \frac{i \cos (b) \sin (a)}{d} + \frac{i \sin (b) \cos (a)}{g} \\ i g \sin (b) \cos (a) + i d \cos (b) \sin (a) & g \frac{\sin (b) (i^{2} \sin (a))}{d} + \cos (b) \cos (a) \end{array}]$

Paso 5.3.1.7

Combina $g$ y $\frac{\sin (b) (i^{2} \sin (a))}{d}$ .

$[\begin{array}{c} \cos (b) \cos (a) - \frac{d \sin (a) \sin (b)}{g} & \frac{i \cos (b) \sin (a)}{d} + \frac{i \sin (b) \cos (a)}{g} \\ i g \sin (b) \cos (a) + i d \cos (b) \sin (a) & \frac{g (\sin (b) (i^{2} \sin (a)))}{d} + \cos (b) \cos (a) \end{array}]$

Paso 5.3.2

Simplifica el numerador.

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Paso 5.3.2.1

Reescribe $i^{2}$ como $- 1$ .

$[\begin{array}{c} \cos (b) \cos (a) - \frac{d \sin (a) \sin (b)}{g} & \frac{i \cos (b) \sin (a)}{d} + \frac{i \sin (b) \cos (a)}{g} \\ i g \sin (b) \cos (a) + i d \cos (b) \sin (a) & \frac{g \sin (b) \cdot - 1 \sin (a)}{d} + \cos (b) \cos (a) \end{array}]$

Paso 5.3.2.2

Factoriza el negativo.

$[\begin{array}{c} \cos (b) \cos (a) - \frac{d \sin (a) \sin (b)}{g} & \frac{i \cos (b) \sin (a)}{d} + \frac{i \sin (b) \cos (a)}{g} \\ i g \sin (b) \cos (a) + i d \cos (b) \sin (a) & \frac{- g \sin (b) \sin (a)}{d} + \cos (b) \cos (a) \end{array}]$

Paso 5.3.3

Mueve el negativo al frente de la fracción.

$[\begin{array}{c} \cos (b) \cos (a) - \frac{d \sin (a) \sin (b)}{g} & \frac{i \cos (b) \sin (a)}{d} + \frac{i \sin (b) \cos (a)}{g} \\ i g \sin (b) \cos (a) + i d \cos (b) \sin (a) & - \frac{g \sin (b) \sin (a)}{d} + \cos (b) \cos (a) \end{array}]$