Question

If \( A \) is a skew-symmetric matrix of odd order, write the value of \( |A| \).

Hint:

Remember:
Skew-symmetric matrix + odd order $\Rightarrow$ determinant is always zero.

Answer 1

Concept: A square matrix \( A \) is said to be skew-symmetric if: \[ A^T = -A \] Key properties of skew-symmetric matrices: \begin{itemize} \item All diagonal elements are zero. \item The determinant of a skew-symmetric matrix of odd order is always zero. \end{itemize} Step 1: Use determinant property.
For any square matrix, \[ |A^T| = |A| \]
Step 2: Apply skew-symmetric condition.
Since \( A^T = -A \), taking determinants: \[ |A^T| = |-A| \]
Step 3: Use determinant rule.
For an \( n \times n \) matrix: \[ |-A| = (-1)^n |A| \] Thus, \[ |A| = (-1)^n |A| \]
Step 4: Consider odd order.
If \( n \) is odd, then \( (-1)^n = -1 \). So, \[ |A| = -|A| \Rightarrow 2|A| = 0 \Rightarrow |A| = 0 \] Conclusion:
The determinant of a skew-symmetric matrix of odd order is always zero.