This is a cross-post.

Let $k,n$ be natural numbers, $1<k<n$. Suppose we have an "unknown" **invertible** $n \times n$ matrix $A$ over a field of characteristic zero. (we do not know the entries of $A$).

Can we recover all the $k$-minors of $A$ from a fixed*, ordered partial list of them?

Explicitly: We are given the values of $r$ of the minors-- a list of $r$ numbers-- and we are told which number corresponds to which minor. Can we recover the other minors?

*The list should be independent of the matrix $A$.

This question is similar to this one, but not identical to it. Here I am talking about a square matrix.

*Comment:* Knowing of all matrix $k$-minors of $A$ is equivalent to knowing $A$ up to a multiplication by a $k$-th root of unity, since for invertible endomorphisms, $\bigwedge^k A=\bigwedge^k B$ if and only if $A=\lambda B$ where $\lambda^k=1$.

Some non-degeneracy assumptions on $A$ are necessary here: We at least need to assume that $\text{rank}(A)>k$. Otherwise, if $\text{rank}(A)\le k$, then even if we know all the $k$-minors of $A$ except one, we cannot recover the last one.

Indeed, take $A=\pmatrix{D&0\\ 0&0}$ where $D$ is any diagonal matrix of size $k$. The $k$-minor corresponding to the first $k$ rows and columns (which is $\det D$) cannot be recovered from the other $k$-minors (which are zeroes).

(Just thinking out loud.)I feel it might be worth generalizing the question to the case where you delete $k$ columns and $ℓ$ rows, with the convention that the “determinant” of a non-square matrix is simply the wedge product of its columns (say). So one sub-question becomes: how many wedge products of $n-k$ among $n$ column vectors do you need to recover them all? (The Plücker relations certainly have something to say here.) And another: now what if we also delete some entries? $\endgroup$(Another thinking out loud attempt!)The matrix defines a section of $S^{*}\otimes Q$ on $G_s\times G_q$ where $S$ is the universal sub-bundle on the Grassmannian $G_s$ of rank $k$ sub-spaces and $Q$ is the universal quotient bundle on the Grassmannian $G_q$ of rank $k$ quotient spaces. Such a sectionisdetermined by its values at finitely many points. However, you are not asking for the value, rather only the value of its image in $\det(S)^*\otimes\det(Q)$. Moreover, the points are "pre-determined" since you are only using points determined by basis vectors. $\endgroup$