# Trace of matrix

Algebra Level 5

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

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