Let $ M_n$ denote the set of $n\times n$ matrices over some field $\mathbb F$, e.g. $\mathbb Q$. For $A\in M_n$, denote $A_i^j$ the minor of $A$ after removing the $i$-th row and $j$-th column.
Let $A$ be some generic matrix in $M_n$. Take $S\in SL_n$ and let $B=SAS^{-1}$. I am interested in the relationship between $A^{k}_k$ and $B_k^k$, especially their determinant.
Can I find a matrix $L\in M_{n-1}$ such that $L A_k^k L^{-1}=B_k^k$?
Is there any linear algebra trick that can address $\frac{\det A_k^k}{\det B_k^k}$?
Thanks.