matrix properties

Question

If A is a skew-symmetric matrix of odd order, then |A| equals

RankGuru · Accepted Answer

1. **Definition**: For a skew-symmetric matrix $A$, $A^T = -A$.
2. **Apply Determinant**: $|A^T| = |-A|$.
3. **Properties**:
   - $|A^T| = |A|$
   - $|-A| = (-1)^n |A|$, where $n$ is the order.
4. **Constraint of Odd Order**: If $n$ is odd, $(-1)^n = -1$.
   - Thus, $|A| = -|A| \implies 2|A| = 0 \implies |A| = 0$.
5. **Result**: The determinant of an odd-order skew-symmetric matrix is always zero.

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