What is the adjoint of the matrix `({:(cos(-theta)-sin(-theta)),(-sin(-theta)cos(-theta)):})`?

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. `({:(cos theta,-sin theta),(sin theta,cos theta):})`

Explanation

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

### Step 1: Simplify the Matrix
Using the properties of trigonometric functions, we know that:
- \( \cos(-\theta) = \cos(\theta) \)
- \( \sin(-\theta) = -\sin(\theta) \)

Substituting these values into the matrix, we get:
\[ A = \begin{pmatrix} \cos(\theta) & \sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix} \]

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

### Step 3: Apply the Formula
In our case, we have:
- \( a = \cos(\theta) \)
- \( b = \sin(\theta) \)
- \( c = \sin(\theta) \)
- \( d = \cos(\theta) \)

Now substituting these values into the adjoint formula:
\[ \text{adj}(A) = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{pmatrix} \]

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

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