Determinant and permutations from Shilov I have been reading chapter 1 of Shilov's Linear Algebra and get stuck at the part where he mentions determinant and permutation:

I just want to ask a couple questions regarding the part that is circled in red.I understand the inversion part, but I don't know how can one determine the formula for n order determinant by using the formula that is circled in red. How does he derive this formula to begin with?
Take the determinant of a 3x3 matrix:
$
\left( \begin{array}{cc}
a_{11} & a_{21} & a_{31}\\ 
a_{12} & a_{22} & a_{32}\\
a_{13} & a_{23} & a_{33}\\ &
\end{array} \right)
%$
The third order determinant is $a_{11}a_{22}a_{13}+a_{21}a_{32}a_{13}+a_{31}a_{12}a_{23}-a_{31}a_{22}a_{13}-a_{11}a_{32}a_{23}-a_{21}a_{12}a_{33}$.
So the sign is negative for $a_{31}a_{22}a_{13}; a_{11}a_{32}a_{23}; a_{21}a_{12}a_{33}$ is because the inversion for each is only 1?
In another Russian book, the determinant is also defined as permutation as well, this image is taken from "Fundamentals of Linear Algebra and Analytical Geometry" by Bugrov and Nikolsky:

Russian books seem to define determinant early on, while more modern books seem to treat it much later.
 A: Are you confused about the meaning of the $\sum$ sign, as in, what the addends are, or are you confused about the signs?


*

*The sum in Shilov's book is a sum over all sequences $\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right)$ that are permutations (i.e., rearrangements) of the sequence $\left(1,2,\ldots,n\right)$. The sum in Bugrov and Nikolsky's book is a similar sum, except that they denote the sequences by $\left(j_1, j_2, \ldots, j_n\right)$ rather than by $\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right)$.

*The sign in the addend corresponding to a permutation $j$ is a $+$ if $j$ has an even number of inversions, and a $-$ if $j$ has an odd number of inversions.
See Appendix B in Neil Strickland's Linear Mathematics for Applications for a more modern exposition of this subject. Modern texts usually differ from older texts (like the one you are reading) in that


*

*they define permutations as bijective maps from $\left\{1,2,\ldots,n\right\}$ to $\left\{1,2,\ldots,n\right\}$ (as opposed to defining them as sequences of numbers), and

*they usually put stuff under $\sum$ signs that explains what the sum is ranging over (something that older texts don't do for reasons of typographical parsimony).
