Which of the following matrices is both symmetric and skew -symmetric ?

Options

A. Identity matrix
B. Diagonal matrix
C. square matrix
D. Null matrix

Answer

To determine which of the following matrices is both symmetric and skew-symmetric, we need to first understand the definitions of symmetric and skew-symmetric matrices.

### Definitions:
1. **Symmetric Matrix**: A matrix \( A \) is symmetric if \( A^T = A \), where \( A^T \) is the transpose of matrix \( A \).
2. **Skew-Symmetric Matrix**: A matrix \( A \) is skew-symmetric if \( A^T = -A \).

### Step-by-Step Solution:

1. **Identify the Properties**:
- A matrix cannot be both symmetric and skew-symmetric unless it is a special case.
- If a matrix is both symmetric and skew-symmetric, then:
\[ A^T = A \quad \text{(symmetric)} \]
\[ A^T = -A \quad \text{(skew-symmetric)} \]

2. **Combine the Equations**:
- From the two properties, we can set them equal to each other:
\[ A = -A \]
- This implies that:
\[ 2A = 0 \quad \Rightarrow \quad A = 0 \]

3. **Conclusion**:
- The only matrix that satisfies both conditions (symmetric and skew-symmetric) is the **null matrix** (or zero matrix), which is a matrix where all elements are zero.

### Final Answer:
The matrix that is both symmetric and skew-symmetric is the **null matrix**.