If a matrix A is symmetric as well as antisymmetric, then which one of the following is correct ?

Options

A. A is a diagonal matrix
B. A is a null matrix
C. A is a unit matrix
D. A is a triangular matrix

Answer

To solve the problem, we need to analyze the properties of a matrix that is both symmetric and antisymmetric.

### Step-by-Step Solution:

1. **Understanding Symmetric Matrix**:
A matrix \( A \) is symmetric if:
\[ A^T = A \]
This means that the transpose of the matrix is equal to the matrix itself.

2. **Understanding Antisymmetric Matrix**:
A matrix \( A \) is antisymmetric if:
\[ A^T = -A \]
This means that the transpose of the matrix is equal to the negative of the matrix.

3. **Combining Both Properties**:
Given that \( A \) is both symmetric and antisymmetric, we can set the two equations equal to each other:
\[ A = A^T \quad \text{(symmetric)} \]
\[ A = -A^T \quad \text{(antisymmetric)} \]

4. **Setting the Equations Equal**:
From the two properties, we have:
\[ A = -A \]
This implies:
\[ A + A = 0 \quad \Rightarrow \quad 2A = 0 \]

5. **Solving for A**:
Dividing both sides by 2 gives:
\[ A = 0 \]
Therefore, the only matrix that is both symmetric and antisymmetric is the zero matrix.

6. **Conclusion**:
Hence, the correct statement is that \( A \) must be the null matrix (zero matrix).

### Final Answer:
The correct conclusion is that if a matrix \( A \) is both symmetric and antisymmetric, then:
\[ A = 0 \quad \text{(the null matrix)} \]