If the matrix A is both symmetric and skew-symmetric , then

Options

A. A is a diagonal matrix
B. A is a zero matrix
C. A is a square matrix
D. None of these

Answer

To solve the problem, we need to analyze the conditions for a matrix to be both symmetric and skew-symmetric.

### Step-by-Step Solution:

1. **Understanding Symmetric and Skew-Symmetric Matrices**:
- A matrix \( A \) is **symmetric** if \( A = A^T \) (where \( A^T \) is the transpose of \( A \)).
- A matrix \( A \) is **skew-symmetric** if \( A = -A^T \).

2. **Setting Up the Conditions**:
- Since the matrix \( A \) is both symmetric and skew-symmetric, we have:
\[ A = A^T \quad \text{(1)} \]
\[ A = -A^T \quad \text{(2)} \]

3. **Combining the Conditions**:
- From equation (1), we can substitute \( A^T \) in equation (2):
\[ A = -A \]
- This implies:
\[ 2A = 0 \quad \Rightarrow \quad A = 0 \]

4. **Conclusion**:
- The only matrix that satisfies both conditions (symmetric and skew-symmetric) is the zero matrix.
- Therefore, the answer is that \( A \) must be the zero matrix.

### Final Answer:
The matrix \( A \) is the zero matrix.

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