Why the difference between the determinant formulas in Russian and English Wikipedia versions?

Question

The article Determinant in Russian version of the Wikipedia tell us that determinant might be found as:

$$\Delta = \sum^n_{j=1}(-1)^{i+j}a_{ij}M^{-i}_j$$

In the English version, the formula is slightly different:

$$\det(A) = \sum^n_{j=1}(-1)^{i+j}a_{i,j}M_{i,j}$$

Could someone, please, explain why the Russian version's sum takes only j-th minor M's element and has minus i power, but the English version has only a single element (i,j)?

Answer 1

The Russian version does not show $$M_j^{-i} $$ but rather $$\bar M_j^i $$ and even explains

где ${\bar {M}}_{j}^{1}$ — дополнительный минор к элементу $a_{1j}$

which is hence just a different notation for the same thing

Answer 2

The Russian version shows particular cases: $$\Delta = \sum^n_{j=1}(-1)^{1+j}a_{1j}\bar{M}^1_j \ \ - \ \text{expansion by the first row elements}$$ $$\Delta = \sum^n_{i=1}(-1)^{i+1}a_{i1}\bar{M}^i_1 \ \ - \ \text{expansion by the first column elements}$$ And then it gives the general formula of expansion by any row (column): $$\Delta = \sum^n_{j=1}(-1)^{i+j}a_{ij}\bar{M}^i_j \ \ (\text{for a fixed $i$})$$ Also, note it is not a minus $i$ power, but a complementary minor: $\bar{M}^{\ \ i}_{\ \ j}$.