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Let $n$ be a positive integer, and let $1<k<n$. Suppose we have an "unknown" real $n \times n$ matrix $A$. (we do not know the entries of $A$).

Can we recover all the $k$-minors of $A$ from an indexed partial list of them?

For example, suppose that we are given the values of all the $k$-minors of $A$ except one of them -specifically we are given an indexed list of $\binom{n}{k}^2-1$ numbers, and we are told which number corresponds to which minor. Can we recover the last minor?

I am interested to know what is the minimal number of $k$-minors needed to recover the rest, in general.

Edit:

It seems that in general we cannot do anything. We probably need some additional assumptions: Perhaps that $A$ is invertible, or at least $\text{rank}(A)>k$.

Indeed, if $\text{rank}(A)\le k$, then we can take $A=\pmatrix{D&0\\ 0&0}$ where $D$ is any diagonal matrix of size $k$. Then we cannot recover the $k$-minor corresponding to the first $k$ rows and columns which is $\det D$ from the other $k$-minors (which are zeroes). This example was suggested by user1551,

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    $\begingroup$ This example may be of interest: for $A=\pmatrix{D&0\\ 0&0}$, where $D$ is any diagonal matrix of size $k=n-1$. We cannot recover the $(n,n)$-th $k$-minor (i.e. $\det D$) from the others (which are zeroes). $\endgroup$ – user1551 Jun 26 at 9:02
  • $\begingroup$ Thank you! This is a nice observation. I have updated the question accordingly. $\endgroup$ – Asaf Shachar Jun 26 at 9:20

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