Without expanding the determinant.

Question

In the below determinant, without expanding the determinant we have to prove that the determinant is zero.

I know how to solve it by expanding, but without that I have no idea.

Answer 1

Hint. The determinant of a skew-symmetric matrix $A$ of odd order is always $0$. Check this by $A=-A^t$ and $\det A^t=\det A$.

Answer 2

The matrix has a null space containing $\begin{pmatrix}a\\ c\\b\end{pmatrix}$ at least. As long as one of $a,b,c$ are nonzero then the nullspace is nontrivial, but if they're all $0$ then the matrix is the $0$ matrix to begin with. Either way, the determinant is $0$ because it has a nontrivial nullspace.