How can I take the determinant on both sides in such a matrix product?

Question

If $A^T BA = O,$ where $A$ is $3\times1$ matrix, then the value of $|B|$ is equal to :

(A) $1$

(B) $–1$

(C) $3$

(D) $0$

I know that the determinant of a non-square matrix is not defined, so here the matrix $B$ has to be a square matrix to be conformable for matrix multiplication. How do I conclude about its determinant in this case?

Answer 1

This is not sufficient information to conclude the determinant of $B$, as we may have $A=0$.

Answer 2

Try $B=\begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & b\end{bmatrix}$ with $b=0$ and with $b\ne 0$. What nonzero vector can you choose for $A$ in either case?

P.S. This is a terrible question.