Cayley-Hamilton Theorem for Mixed Discriminants

The mixed discriminant of an \(n\)-tuple of \(n \times n\) matrices \(A_1, \ldots, A_n\) is defined as $$\mathcal{D}(A_1, A_2, \ldots, A_n) = \frac{1}{n!} \sum_{\sigma \in S(n)} \det(A_{\sigma(1)}^{(1)}, A_{\sigma(2)}^{(2)}, \ldots, A_{\sigma(n)}^{(n)}),$$
where \(A^{(i)}\) denotes the \(i\)th column of the matrix \(A\) and \(S(n)\) denotes the group of permutations of \(1, 2, \ldots, n\). For \(n\) matrices \(A_1, \ldots, A_n\) and indeterminates \(\lambda_1, \ldots, \lambda_n\), set $$\Phi_{\lambda_1, \ldots, \lambda_n}(A_1, \ldots, A_n) = \mathcal{D}(\lambda_1 I – A_1, \ldots, \lambda_n I – A_n).$$
It is shown that \(\Phi_{A_1, \ldots, A_n}(A_1, \ldots, A_n) = 0\).