The multiplicative inverse of `A=[(cos theta,-sin theta),(sin theta, cos theta)]` is equal to

Options

A. `[(-cos theta, sin theta),(-sin theta, -cos theta)]`
B. `[(cos theta, sin theta),(-sin theta, cos theta)]`
C. `[(-cos theta, -sin theta),(sin theta, -cos theta)]`
D. `[(costheta, sin theta),(sin theta,-cos theta)]`

Answer

To find the multiplicative inverse of the matrix \( A = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant of Matrix A
The determinant of a \( 2 \times 2 \) matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:
\[ \text{det}(A) = ad - bc \]
For our matrix \( A \):
- \( a = \cos \theta \)
- \( b = -\sin \theta \)
- \( c = \sin \theta \)
- \( d = \cos \theta \)

Thus, the determinant is:
\[ \text{det}(A) = (\cos \theta)(\cos \theta) - (-\sin \theta)(\sin \theta) = \cos^2 \theta + \sin^2 \theta \]
Using the Pythagorean identity, we know that:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]
So, \( \text{det}(A) = 1 \).

### Step 2: Find the Adjoint of Matrix A
The adjoint of a \( 2 \times 2 \) matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:
\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]
For our matrix \( A \):
- \( d = \cos \theta \)
- \( -b = \sin \theta \)
- \( -c = -\sin \theta \)
- \( a = \cos \theta \)

Thus, the adjoint is:
\[ \text{adj}(A) = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \]

### Step 3: Calculate the Inverse of Matrix A
The inverse of a matrix \( A \) can be calculated using the formula:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
Since we found that \( \text{det}(A) = 1 \), we have:
\[ A^{-1} = \frac{1}{1} \cdot \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \]

### Final Answer
Thus, the multiplicative inverse of the matrix \( A \) is:
\[ A^{-1} = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \]