I am currently going through Linear Algebra by Shilov and I am having some trouble understanding his derivation of the formula of a determinant. He first introduces the product
\begin{equation} a_{{\alpha_1}1}a_{{\alpha_2}2}\cdots a_{{\alpha_n}n} \tag{1} \end{equation}
which I am assuming is the product of the principle diagonal. From what I understand $\alpha_1$ is the row and the superscript to $\alpha_1$ is the column. Then he goes onto explain the inversion (denoted by $N(\cdots)$) of the sequence of $(\alpha_1, \alpha_2, \cdots, \alpha_n)$ which is when we put a larger indice before a smaller one. For example, $N(4,3,1,2)$ is equal to $5$. Then he explains that if the answer is an even number then make it positive and if it's an odd number than the product is negative. I understand everything up to this point (and if someone could verify my understanding up to this point, that would be awesome).
Then he goes onto explain that the total number of products we can form are $n!$ of $(1)$ and this is the part where I get confused. When we try to find the determinant of, say, a $2 \times 2$ matrix, we have two permutations namely $1,2$ and $2,1$ so our product should be $a_{1,i}\cdot a_{2,i}$ and $a_{2,i} \cdot a_{1,i}$. However, I am having trouble understanding how we find the $i$. He lists the formula for a $2 \times 2$ matrix as $a_{11}\cdot a_{22} - a_{21}\cdot a_{12}$ but I have no idea how he arrives at the various column indexes.
Can someone please point me in the correct direction or explain where I am going wrong?
Thanks a lot for your time
EDIT:
According to the comments, my understanding of $\alpha_1$ is incorrect. Suppose, we have a random matrix then is $\alpha_1$ the first row? And $a_{{\alpha_1}1}$ the boxed entry? \begin{pmatrix} \boxed{1} & a_1 & a_1^2 & \cdots & a_1^n \\ 1 & a_2 & a_2^2 & \cdots & a_2^n \\ \vdots & \vdots& \vdots & \ddots & \vdots \\ 1 & a_m & a_m^2 & \cdots & a_m^n \end{pmatrix}