Let \(A = \begin{bmatrix} 0 & 1 & -1 \\ 1 & 3 &2 \\ 4 & 1 &-2\end{bmatrix}\). Let \(\sigma \in S_3\) be the permutation \(\left(\begin{matrix} 1 & 2 & 3\\ 2 & 3 & 1\end{matrix}\right)\).
What is the value of \(A_{1,\sigma(1)} A_{2,\sigma(2)} A_{3,\sigma(3)}\)?