If a square matrix A is orthogonal as well as symmetric, then

Options

A. A is involutory matrix
B. A is idempotent matrix
C. A is a diagonal matrix
D. none of these

Answer

To solve the problem, we need to analyze the properties of the square matrix \( A \) given that it is both orthogonal and symmetric.

### Step-by-Step Solution:

1. **Understanding Orthogonal Matrix**:
A matrix \( A \) is orthogonal if \( A^T = A^{-1} \). This means that the transpose of \( A \) is equal to its inverse.

2. **Understanding Symmetric Matrix**:
A matrix \( A \) is symmetric if \( A = A^T \). This means that the matrix is equal to its transpose.

3. **Combining Properties**:
Since \( A \) is both orthogonal and symmetric, we can combine the two properties:
\[ A = A^T \quad \text{and} \quad A^T = A^{-1} \]
Therefore, we can conclude:
\[ A = A^{-1} \]

4. **Multiplying Both Sides by \( A \)**:
If we multiply both sides of the equation \( A = A^{-1} \) by \( A \), we get:
\[ A \cdot A = A \cdot A^{-1} \]
This simplifies to:
\[ A^2 = I \]
where \( I \) is the identity matrix.

5. **Conclusion**:
The equation \( A^2 = I \) indicates that \( A \) is an involutory matrix. An involutory matrix is defined as a matrix that, when multiplied by itself, yields the identity matrix.

### Final Answer:
Thus, if a square matrix \( A \) is orthogonal as well as symmetric, then \( A \) is an involutory matrix.