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If two indices of the n-dimensional Levi-Civita symbol are equal, exchanging them leaves the component unchanged. Antisymmetry of a multilinear map simultaneously negates it, so $ϵ=-ϵ$. Over the real numbers this implies $\boxed{ϵ=0}$.

Let $M$ be the square matrix whose $j$th column is $A_j$. The Leibniz formula for determinants is precisely the stated contraction:A square matrix has zero determinant exactly when its columns are linearly dependent, proving the required equivalence. One direction also follows directly from multilinearity and antisymmetry of a multilinear map: replacing one column by a linear combination of the others leaves only terms with repeated columns, each zero. Conversely, nonzero determinant gives an inverse matrix and hence independent columns.