I am reading Shilov's book linear algebra. He explains how to compute determinants. Basically, for the plus terms you write \begin{equation} x_{a1}x_{b2}x_{c3}x_{d4}x_{e5} x_{f6} \end{equation} and then permute the left side indices, giving \begin{align} &x_{a1}x_{b2}x_{c3}x_{d4}x_{e5} x_{f6}\\ &x_{b1}x_{c2}x_{d3}x_{r4}x_{f5} x_{a6}\\ &x_{c1}x_{d2}x_{e3}x_{f4}x_{a5} x_{b6}\\ &x_{d1}x_{e2}x_{f3}x_{a4}x_{b5} x_{c6}\\ &x_{e1}x_{f2}x_{a3}x_{b4}x_{c5} x_{d6}\\ &x_{f1}x_{a2}x_{b3}x_{c4}x_{d5} x_{e6}\\ \end{align} I just used letters to make it obvious which index was permuting. You should think of it like $a=1,b=2$, etc
For the minus terms you write
\begin{equation} x_{f1}x_{e2}x_{d3}x_{c4}x_{b5} x_{a6} \end{equation}
and do the same thing.
I understood all of that. Made sense.
My Confusion
I want to find the sign of \begin{equation} a_{23}a_{31}a_{42}a_{56}a_{14} a_{65} \end{equation} for this term that appears in a determinant of order $6$.
But if I rearrange to make the right indices in order,
\begin{equation} a_{31}a_{42}a_{23}a_{14} a_{65}a_{56} \end{equation}
you can see the left indices read \begin{equation} 342165 \end{equation} which isn't in order. \begin{equation} 165342 \end{equation}
The above determinant rules only work if the left hand indices are in numerical order like \begin{equation} 432165 \end{equation} My Question:
How can I compute a determinant if the left hand indices aren't in order?