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 determine what happens when a matrix \( A \) is both symmetric and skew-symmetric, we will analyze the definitions of these types of matrices and derive the conclusion step by step.

### Step-by-Step Solution

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

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

3. **Given Condition**:
We know that \( A \) is both symmetric and skew-symmetric. Therefore, we can write:
\[ A = A^T \quad \text{(symmetric)} \]
\[ A = -A^T \quad \text{(skew-symmetric)} \]

4. **Equating the Two Conditions**:
Since both conditions must hold true, we can set them equal to each other:
\[ A = A^T = -A^T \]

5. **Adding \( A^T \) to Both Sides**:
If we add \( A^T \) to both sides of the equation \( A = -A^T \), we get:
\[ A + A^T = 0 \]
Since \( A = A^T \), we can substitute \( A \) for \( A^T \):
\[ A + A = 0 \implies 2A = 0 \]

6. **Conclusion**:
Dividing both sides by 2 gives us:
\[ A = 0 \]
Therefore, if \( A \) is both symmetric and skew-symmetric, then \( A \) must be the zero matrix.

### Final Answer:
If \( A \) is a matrix that is both symmetric and skew-symmetric, then \( A \) is the zero matrix.

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