If A = `((costheta,-sintheta),(sintheta,costheta))` then

Options

A. A is an orthogonal matrix
B. A is a symmetric matrix
C. A is a skew -symmetric matrix
D. none of these

Answer

To solve the problem, we need to analyze the matrix \( A \) given by:

\[ A = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \]

We will check if \( A \) is an orthogonal matrix, symmetric matrix, or skew-symmetric matrix.

### Step 1: Check if \( A \) is an orthogonal matrix

An orthogonal matrix \( A \) satisfies the condition:

\[ A^T A = I \]

where \( A^T \) is the transpose of \( A \) and \( I \) is the identity matrix.

#### Calculate \( A^T \):

\[ A^T = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \]

#### Calculate \( A^T A \):

\[ A^T A = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \]

Calculating the product:

1. First row, first column:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]

2. First row, second column:
\[ \cos \theta (-\sin \theta) + \sin \theta \cos \theta = 0 \]

3. Second row, first column:
\[ -\sin \theta \cos \theta + \cos \theta \sin \theta = 0 \]

4. Second row, second column:
\[ -\sin^2 \theta + \cos^2 \theta = 1 \]

Thus, we have:

\[ A^T A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I \]

Since \( A^T A = I \), \( A \) is an orthogonal matrix.

### Step 2: Check if \( A \) is symmetric

A matrix \( A \) is symmetric if:

\[ A^T = A \]

We already calculated \( A^T \):

\[ A^T = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \]

Since \( A \neq A^T \) (specifically, the off-diagonal elements differ), \( A \) is not symmetric.

### Step 3: Check if \( A \) is skew-symmetric

A matrix \( A \) is skew-symmetric if:

\[ A^T = -A \]

Calculating \( -A \):

\[ -A = \begin{pmatrix} -\cos \theta & \sin \theta \\ -\sin \theta & -\cos \theta \end{pmatrix} \]

Since \( A^T \neq -A \) (the diagonal elements differ), \( A \) is not skew-symmetric.

### Conclusion

- \( A \) is an orthogonal matrix.
- \( A \) is not symmetric.
- \( A \) is not skew-symmetric.

### Final Answer

The correct option is that \( A \) is an orthogonal matrix.

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