I prefer to start with an example. Consider the matrix $$ A = \left[ \begin{array}{ccc} 1 & 1 & 0 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \end{array} \right] $$ It is invertible, since its determinant is $1$. Now consider the sequence of the upper-left square submatrices:

$$ A \left[ 1 \right] = \left[ \begin{array}{c} 1 \end{array} \right], \quad A \left[ 1,2 \right] = \left[ \begin{array}{cc} 1 & 1\\ 1 & 1 \end{array} \right], \quad A \left[ 1,2,3 \right] = A $$ The second matrix is singular (its determinant is zero). However, we can find other sequence of nested "central" sub-matrices such that all of them are invertible: $$ A \left[ 1 \right] = \left[ \begin{array}{c} 1 \end{array} \right], \quad A \left[ 1,3 \right] = \left[ \begin{array}{cc} 1 & 0\\ 1 & 1 \end{array} \right], \quad A \left[ 1,2,3 \right] = A $$ I wonder when such a sequence exists. I need some notation to enunciate the problem.

Let $A$ be a $n \times n$ matrix. Given $S \subset \left[ 1,n \right]$, we denote $A\left[ S \right]$ the matrix obtained from $A$ by only taking the rows and columns whose index are in $S$. Formally, $$ A\left[ S \right] = R_S \, A \, R^t_S, $$ where $S = \lbrace s_1 < \ldots < s_m \rbrace$ and $R_s$ is the $m \times n$ matrix with $1$ at the positions $(i, s_i)$ and zero elsewhere.

Question: Let $A$ be a regular $n \times n$ matrix. Under which conditions there exist a nested sequence $$ S_1 \subset S_2 \subset \cdots \subset S_n = \left[ 1,n \right], \quad |S_i| = i$$ such that each $A\left[ S_i \right]$ is regular? How to compute such a sequence? How many sequences are there?

It is trivial that the matrix must have a non-zero entry in its diagonal. I am particularly interested in the case where all the entries in the diagonal are non-zero.

Does anybody know if this problem has already been studied?

Problem: Let $A$ be a $n \times n$ integer matrix, $\det(A) = \pm 1$. Under which conditions there exist a nested sequence of principal submatrices of size $n$ such that they all have determinant $\pm 1$ ?

We call such a sequence a $\pm 1$ Principal Minor Sequence ($\pm 1$PMS). Note that such a sequence is described by a permutation of $n$ elements. For example, consider the matrix $$ A = \left[ \begin{array}{ccc} 1 & 1 & 0 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \end{array} \right]. $$ The sequence $(1,2,3)$ $$ A \left[ 1 \right] = \left[ \begin{array}{c} 1 \end{array} \right], \quad A \left[ 1,2 \right] = \left[ \begin{array}{cc} 1 & 1\\ 1 & 1 \end{array} \right], \quad A \left[ 1,2,3 \right] = A. $$ is not a $\pm 1$PMS since $\det(A \left[ 1,2 \right])=0$. The sequence $(1,3,2)$ is a $\pm 1$PMS. On the other hand, the matrix $$ A = \left[ \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 0 & 1 \\ 1 & 0 & 1 \end{array} \right] $$ does not have any $\pm 1$PMS.

Does anybody know if this problem has already been studied? A related problem is to estimate the number of $\pm 1$PMS of a matrix.

Note: It is trivial that a matrix has no $\pm 1$PMS if all its entries in the diagonal are zero.