Trace of matrix

Let $A , B \in M_{2017} (\mathbb R)$ and $X$ a non-zero vector of $\mathbb R^{2017}$ . We assume that $AX=0$ and there is a vector $Y \in \mathbb R^{2017}$ so that $AY=BX$ . Let $A_{j}$ a matrix obtained by replacing the $j^\text{th}$ column of $A$ with that of $B$ .

Calculate $\sum_{j=1}^{2017} \det(A_j).$