A matrix which is both symmetric and skew-symmetric is a

Options

A. Unit matrix
B. Null matrix
C. Non-singular matrix
D. None of these

Answer

To solve the problem, we need to determine the nature of a matrix that is both symmetric and skew-symmetric.

### Step-by-Step Solution:

1. **Definition of Symmetric Matrix**:
A matrix \( A \) is symmetric if:
\[ A = A^T \]
This means that the element at the \( i^{th} \) row and \( j^{th} \) column is equal to the element at the \( j^{th} \) row and \( i^{th} \) column, i.e.,
\[ A_{ij} = A_{ji} \quad \text{for all } i, j. \]

**Hint**: Recall that symmetric matrices have mirrored elements across the main diagonal.

2. **Definition of Skew-Symmetric Matrix**:
A matrix \( A \) is skew-symmetric if:
\[ A = -A^T \]
This implies that the element at the \( i^{th} \) row and \( j^{th} \) column is the negative of the element at the \( j^{th} \) row and \( i^{th} \) column, i.e.,
\[ A_{ij} = -A_{ji} \quad \text{for all } i, j. \]

**Hint**: Remember that skew-symmetric matrices have elements that are negatives of each other across the main diagonal.

3. **Combining the Definitions**:
Since the matrix \( A \) is both symmetric and skew-symmetric, we can set up the following equations based on the definitions:
- From symmetry:
\[ A_{ij} = A_{ji} \]
- From skew-symmetry:
\[ A_{ij} = -A_{ji} \]

4. **Equating the Two Definitions**:
From the two equations above, we have:
\[ A_{ij} = A_{ji} \quad \text{and} \quad A_{ij} = -A_{ji} \]
This implies:
\[ A_{ij} = -A_{ij} \]

**Hint**: Think about what it means for a number to be equal to its own negative.

5. **Solving the Equation**:
From the equation \( A_{ij} = -A_{ij} \), we can add \( A_{ij} \) to both sides:
\[ A_{ij} + A_{ij} = 0 \implies 2A_{ij} = 0 \implies A_{ij} = 0 \]
This means that every element of the matrix \( A \) is zero.

**Hint**: If every entry of a matrix is zero, what type of matrix do you have?

6. **Conclusion**:
Since all entries \( A_{ij} \) are zero for all \( i \) and \( j \), the matrix \( A \) is the zero matrix, also known as the null matrix.

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