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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