If `A` is a skew-symmetric matrix and `n` is odd positive integer, then `A^n` is

Options

A. a skew symmertric matrix
B. a symmetric matrix
C. a zero matrix
D. None of these

Explanation

To solve the problem, we need to determine the nature of \( A^n \) where \( A \) is a skew-symmetric matrix and \( n \) is an odd positive integer.

### Step-by-Step Solution:

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

**Hint**: Remember that for a skew-symmetric matrix, the transpose is equal to the negative of the matrix itself.

2. **Properties of Transpose**:
For any matrix \( A \), the transpose of a product of matrices is given by \( (AB)^T = B^T A^T \). This property will be useful in our calculations.

**Hint**: Keep in mind how transposes work when dealing with matrix products.

3. **Calculating \( A^n \)**:
We want to find \( A^n \) where \( n \) is an odd positive integer. We can express this as:
\[ A^n = A \cdot A \cdot A \cdots A \quad (n \text{ times}) \]

4. **Using the Skew-Symmetry Property**:
Since \( A \) is skew-symmetric, we have:
\[ A^T = -A \]

5. **Finding the Transpose of \( A^n \)**:
We can find the transpose of \( A^n \):
\[ (A^n)^T = (A \cdot A \cdots A)^T = A^T \cdot A^T \cdots A^T = (-A) \cdot (-A) \cdots (-A) = (-1)^n A^n \]

6. **Substituting \( n \)**:
Since \( n \) is an odd positive integer, \( (-1)^n = -1 \). Therefore:
\[ (A^n)^T = -A^n \]

7. **Conclusion**:
This shows that \( A^n \) is also a skew-symmetric matrix because it satisfies the condition \( (A^n)^T = -A^n \).

**Final Answer**: \( A^n \) is a skew-symmetric matrix.