I am struggling to derive the Jacobi Formula for determinants using a method of my own:
Since $det(A)$ is a function of the elements of $(A_{ij})$, write $$det(A)=F(A_{kr}); k,r=0,1,...,N$$ so the differential of the determinant is $$ddet(A)=\sum_{kr}\frac{\partial det(A)}{\partial A_{kr}} dA_{kr} \tag{1}$$. From the Laplacian Expansion of a determinant: $$det(A)=\sum_{j}A_{ij}C^{ij} \tag{2}$$ where $C^{ij}$ is the cofactor in $A_{ij}$ Evaluating the partial derivative in (1) using (2): $$ddet(A)=\sum_{jkr}\left[\frac{ \partial A_{ij}}{ \partial A_{kr}}C^{ij}+A_{ij}\frac{\partial C^{ij}}{\partial A_{kr}}\right]dA_{kr}=\sum_{jkr}\left[\underbrace{\delta _i ^k}_{\text{since is fixed}}\delta _j^r C^{ij}+A_{ij}\frac{\partial C^{ij}}{\partial A_{kr}}\right]dA_{kr}$$ $$=\underbrace{\sum_{j}C^{ij}dA_{ij}}_{\text{leads to the Jacobi Formula}}+ \underbrace{\sum_{jkr}A_{ij}\frac{\partial C^{ij}}{\partial A_{kr}}dA_{kr}}_{\text{but this term is non-zero except for k=i or j=r?}} \tag{3}$$
However on Wikipedia https://en.wikipedia.org/wiki/Jacobi%27s_formula the derivation using the Laplace Expansion involves: $$\sum_{jr}\frac{\partial A_{ij}}{\partial A_{ir}}C^{ij}+ \underbrace{\sum_{jr} A_{ij}\frac{\partial C^{ij}}{\partial A_{ir}}}_{\text{which is indeed zero}} \tag{4}$$
But doesn't the use of (3) assume that $$ddet(A)=\sum_{j}\frac{\partial det(A)}{\partial A_{ij}}dA_{ij}$$ i.e. no sum over $i$ in which case the differential is incomplete?
What did do I wrong and why is my interpretation of the Wikipedia proof incorrect?
Recall what is $C^{ij}$. It is independent of $A_{ir}$ for all $r$, since it is an minor of the $i$th row. More precisely, $$ C^{ij}=G(\{X_{rt}\}_{r\neq i,t\neq j}) $$ for some polynomial function $G$ of $(n-1)^2$ variables. So for all $i,j\leq n$, $$ \frac{\partial C^{ij}}{\partial X_{ir}}=0,\qquad r=1,...,n. $$