Let K = 0, 2, -3; -2, 0, 5; 3, -5, 0. Use |K^T| = |-K| to explain why |K| = 0, and check it by…

Step 1. The argument. $|K| = |K^T| = |-K| = (-1)^3|K| = -|K|$. So $2|K| = 0$ and $|K| = 0$. This works for every skew-symmetric matrix of odd order.

Step 2. Check by expanding. Along row $1$: $0 - 2(0 - 15) + (-3)(10 - 0) = 30 - 30 = 0$.

Step 3. The multiple. $|3K| = 3^3 \times 0 = 0$.