A matrix which is both symmetric as well as skew-symmetric is

Options

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

Explanation

To solve the problem of determining what type of matrix is both symmetric and skew-symmetric, we will follow these steps:

### Step-by-Step Solution:

1. **Definition of Symmetric Matrix**:
A matrix \( A \) is symmetric if \( A^T = A \), where \( A^T \) is the transpose of matrix \( A \).

2. **Definition of Skew-Symmetric Matrix**:
A matrix \( A \) is skew-symmetric if \( A^T = -A \).

3. **Equating the Definitions**:
If a matrix \( A \) is both symmetric and skew-symmetric, we can write:
\[ A^T = A \quad \text{(from symmetry)} \]
\[ A^T = -A \quad \text{(from skew-symmetry)} \]

4. **Setting the Equations Equal**:
Since both expressions equal \( A^T \), we can set them equal to each other:
\[ A = -A \]

5. **Solving the Equation**:
Adding \( A \) to both sides gives:
\[ A + A = 0 \implies 2A = 0 \]
Dividing both sides by 2 results in:
\[ A = 0 \]

6. **Conclusion**:
The only matrix that satisfies both conditions (symmetric and skew-symmetric) is the zero matrix. Therefore, the answer is that a matrix which is both symmetric and skew-symmetric is the zero matrix.

### Final Answer:
The matrix which is both symmetric as well as skew-symmetric is the **zero matrix**.

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