Which of the following the following is correct?

Options

A. skew symmetric matrix of even order is always singular
B. skew symmetric matrix of odd order is non singular
C. skew symmetric matrix of odd order is singular
D. None of these

Answer

To determine which of the statements regarding skew-symmetric matrices is correct, let's analyze the properties of skew-symmetric matrices.

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

### Step 2: Properties of Determinants
One important property of determinants is that if a matrix is skew-symmetric and of odd order (i.e., the number of rows or columns is odd), then its determinant is zero. This means that such a matrix is singular.

### Step 3: Analyzing the Options
1. **Option 1**: "Skew symmetric matrix of even order is always singular."
- This is **not necessarily true**. A skew-symmetric matrix of even order can be non-singular.

2. **Option 2**: "Skew symmetric matrix of odd order is non-singular."
- This is **false**. As established, a skew-symmetric matrix of odd order is singular (determinant = 0).

3. **Option 3**: "Skew symmetric matrix of odd order is singular."
- This is **true**. A skew-symmetric matrix of odd order has a determinant of zero.

4. **Option 4**: "Skew symmetric matrix of even order is non-singular."
- This is **not necessarily true**. While some even-order skew-symmetric matrices can be non-singular, others can be singular.

### Conclusion
The correct statement is:
- **Skew symmetric matrix of odd order is singular.**