I have a matrix $A$ of order $d$, and its inverse $A^{-1}$. I am trying to find out whether there is some kind of relationship between submatrices of the two. Is there any relationship regarding the invertibility of submatrices of $A$ and $A^{-1}$? Maybe something about the rank of such submatrices.
For example, how true is the following statement: if there is a submatrix of order $c$ of $A$ that has full rank then there is a submatrix of of $A^{-1}$ of order $d-c$ that is also full rank.
I think this answers the question: if a $d\times d$ matrix $A$ is invertible, then it has $c\times c$ submatrices of full rank for every $c$, $1\le c\le d$.
The proof is quite simple. By using expansion along a row or column, repeatedly if necessary, the determinant of $A$ can be expressed as linear combination of determinants of $c\times c$ matrices. If there were a value of $c$ for which no $c\times c$ submatrix had full rank, then all those submatrices would have determinant zero, and then $A$ would have determinant zero, contradicting its invertibility.