The inverse of skew - symmetric matrix of odd order

Options

A. is a symmetric matrix
B. is a diogonal matrix
C. is a skew - symmetric matrix
D. does not exist

Answer

To determine the inverse of a skew-symmetric matrix of odd order, we can follow these steps:

### Step-by-Step Solution:

1. **Definition of Skew-Symmetric Matrix**:
A matrix \( A \) is called skew-symmetric if \( A^T = -A \). This means that the transpose of the matrix is equal to the negative of the matrix itself.

2. **Properties of Determinants**:
For any skew-symmetric matrix of odd order \( n \), it is known that the determinant of the matrix is zero. This is a fundamental property of skew-symmetric matrices.

3. **Determinant of Skew-Symmetric Matrix**:
Let \( A \) be a skew-symmetric matrix of odd order \( n \). Then:
\[ \text{det}(A) = 0 \]

4. **Inverse of a Matrix**:
The inverse of a matrix \( A \) can be expressed as:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
where \( \text{adj}(A) \) is the adjugate of \( A \).

5. **Substituting the Determinant**:
Since we have established that \( \text{det}(A) = 0 \) for a skew-symmetric matrix of odd order, substituting this into the formula for the inverse gives:
\[ A^{-1} = \frac{1}{0} \cdot \text{adj}(A) \]
This expression is undefined.

6. **Conclusion**:
Therefore, the inverse of a skew-symmetric matrix of odd order does not exist.

### Final Answer:
The inverse of a skew-symmetric matrix of odd order does not exist.