If P is a two-rowed matrix satisfying `P' = P^(-1)`, then P can be

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. none of these

Answer

To solve the problem, we need to analyze the condition given for the matrix \( P \). The condition states that \( P' = P^{-1} \), where \( P' \) denotes the transpose of the matrix \( P \).

### Step-by-Step Solution:

1. **Understanding the Condition**:
Given \( P' = P^{-1} \), we can rewrite this as:
\[ P^T = P^{-1} \]
This means that the transpose of matrix \( P \) is equal to its inverse.

2. **Multiplying Both Sides**:
If we multiply both sides of the equation by \( P \) from the right, we get:
\[ P^T P = I \]
where \( I \) is the identity matrix. This indicates that \( P \) is an orthogonal matrix.

3. **Properties of Orthogonal Matrices**:
An orthogonal matrix has the property that its rows (and columns) are orthonormal vectors. This means:
- The dot product of each row with itself is 1.
- The dot product of different rows is 0.

4. **Form of the Matrix**:
A general 2x2 orthogonal matrix can be expressed in the form:
\[ P = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \]
for some angle \( \theta \). This matrix represents a rotation in the plane.

5. **Verification**:
To verify that this matrix satisfies the condition \( P^T = P^{-1} \):
- Calculate \( P^T \):
\[ P^T = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \]
- Calculate \( P^{-1} \):
The inverse of \( P \) is given by:
\[ P^{-1} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}^T = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \]
- Since \( P^T = P^{-1} \), the condition is satisfied.

6. **Conclusion**:
Therefore, the matrix \( P \) can indeed be represented as:
\[ P = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \]
for any angle \( \theta \).