# How many matrix minors determine all the minors?

Let $$n$$ be a positive integer, and let $$1. Suppose we have an "unknown" real $$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,

• 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). – user1551 Jun 26 at 9:02
• Thank you! This is a nice observation. I have updated the question accordingly. – Asaf Shachar Jun 26 at 9:20