What is the value of the determinant of a skew-symmetric matrix of order $3$?

Solution

1. Let $A$ be a skew-symmetric matrix, so $A^T=-A$.
2. Taking the determinant on both sides, $\det (A^T)=\det (-A)$.
3. Using properties, $\det (A)=(-1{)}^n\det (A)$ where $n$ is the order.
4. For $n=3$, $\det (A)=-\det (A)$, which implies $2\det (A)=0$.
5. Hence the answer is (D).