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Question: The multiplicative inverse of A =\(\begin{bmatrix} \cos\theta & - \sin\theta \\ \sin\theta & \cos\th...

The multiplicative inverse of A =[cos⁡θ−sin⁡θsin⁡θcos⁡θ]\begin{bmatrix} \cos\theta & - \sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}is

A

$\begin{bmatrix}

  • \cos\theta & \sin\theta \
  • \sin\theta & - \cos\theta \end{bmatrix}$
B

$\begin{bmatrix} \cos\theta & \sin\theta \

  • \sin\theta & \cos\theta \end{bmatrix}$
C

$\begin{bmatrix}

  • \cos\theta & - \sin\theta \ \sin\theta & - \cos\theta \end{bmatrix}$
D

[cos⁡θsin⁡θsin⁡θ−cos⁡θ]\begin{bmatrix} \cos\theta & \sin\theta \\ \sin\theta & - \cos\theta \end{bmatrix}

Answer

$\begin{bmatrix} \cos\theta & \sin\theta \

  • \sin\theta & \cos\theta \end{bmatrix}$
Explanation

Solution

|A| = cos2q + sin2 q = 1

Adj. A =$\begin{bmatrix} \cos\theta & \sin\theta \

  • \sin\theta & \cos\theta \end{bmatrix}$

A–1=Adj.A∣A∣\frac{Adj.A}{|A|}= $\begin{bmatrix} \cos\theta & \sin\theta \

  • \sin\theta & \cos\theta \end{bmatrix}$