Álgebra linear Exemplos

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

Etapa 1

Combine $\frac{i}{g}$ e $\sin (b)$ .

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

Etapa 2

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 3

Combine $\frac{i}{d}$ e $\sin (a)$ .

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

Etapa 4

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 5

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

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Etapa 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 $2 \times 2$ .

Etapa 5.2

Multiplique cada linha na primeira matriz por cada coluna na segunda matriz.

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

Etapa 5.3

Simplifique cada elemento da matriz multiplicando todas as expressões.

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Etapa 5.3.1

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

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Etapa 5.3.1.1

Combine $i$ e $\frac{i \sin (a)}{d}$ .

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

Etapa 5.3.1.2

Eleve $i$ à potência de $1$ .

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

Etapa 5.3.1.3

Eleve $i$ à potência de $1$ .

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

Etapa 5.3.1.4

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

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

Etapa 5.3.1.5

Some $1$ e $1$ .

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

Etapa 5.3.1.6

Combine $\sin (b)$ e $\frac{i^{2} \sin (a)}{d}$ .

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

Etapa 5.3.1.7

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

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

Etapa 5.3.2

Simplifique o numerador.

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Etapa 5.3.2.1

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

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

Etapa 5.3.2.2

Fatore o negativo.

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

Etapa 5.3.3

Mova o número negativo para a frente da fração.

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