Lineare Algebra Beispiele

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

Schritt 1

Kombiniere $\frac{i}{g}$ und $\sin (b)$ .

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

Schritt 2

Schreibe neu unter Anwendung des Kommutativgesetzes der Multiplikation.

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

Schritt 3

Kombiniere $\frac{i}{d}$ und $\sin (a)$ .

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

Schritt 4

Schreibe neu unter Anwendung des Kommutativgesetzes der Multiplikation.

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

Schritt 5

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

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Schritt 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 $2 \times 2$ .

Schritt 5.2

Multipliziere jede Zeile in der ersten Matrix mit jeder Spalte in der zweiten Matrix.

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

Schritt 5.3

Vereinfache jedes Element der Matrix durch Ausmultiplizieren aller Ausdrücke.

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Schritt 5.3.1

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

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Schritt 5.3.1.1

Kombiniere $i$ und $\frac{i \sin (a)}{d}$ .

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

Schritt 5.3.1.2

Potenziere $i$ mit $1$ .

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

Schritt 5.3.1.3

Potenziere $i$ mit $1$ .

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

Schritt 5.3.1.4

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

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

Schritt 5.3.1.5

Addiere $1$ und $1$ .

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

Schritt 5.3.1.6

Kombiniere $\sin (b)$ und $\frac{i^{2} \sin (a)}{d}$ .

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

Schritt 5.3.1.7

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

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

Schritt 5.3.2

Vereinfache den Zähler.

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Schritt 5.3.2.1

Schreibe $i^{2}$ als $- 1$ um.

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

Schritt 5.3.2.2

Faktorisiere das negative Vorzeichen heraus.

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

Schritt 5.3.3

Ziehe das Minuszeichen vor den Bruch.

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