If A is symmetric as well as skew-symmetric matrix, then A is

Options

A. diagonal matrix
B. null matrix
C. triangular materix
D. none of these

Explanation

To solve the problem, we need to show that if a matrix \( A \) is both symmetric and skew-symmetric, then \( A \) must be the zero matrix (null matrix).

### Step-by-Step Solution:

1. **Definitions**:
- A matrix \( A \) is **symmetric** if \( A = A^T \) (the transpose of \( A \)).
- A matrix \( A \) is **skew-symmetric** if \( A = -A^T \).

2. **Set Up the Equations**:
- Since \( A \) is symmetric, we can write:
\[ A = A^T \]
- Since \( A \) is skew-symmetric, we can write:
\[ A = -A^T \]

3. **Combine the Equations**:
- From the two equations, we have:
\[ A = A^T \quad \text{(1)} \]
\[ A = -A^T \quad \text{(2)} \]
- Now, we can equate the two expressions for \( A \):
\[ A = -A \quad \text{(substituting from (2) into (1))} \]

4. **Simplify the Equation**:
- Adding \( A \) to both sides gives:
\[ A + A = 0 \]
\[ 2A = 0 \]

5. **Solve for \( A \)**:
- Dividing both sides by 2, we find:
\[ A = 0 \]

6. **Conclusion**:
- Therefore, if \( A \) is both symmetric and skew-symmetric, it must be the zero matrix:
\[ A = \text{Null Matrix} \]