Disclaimer: This question has been moved from MO, as SE seems to be a more appropriate venue for it.
Let $n$ be a large positive integer.
Question
What is a simple way to construct (or characterize) real $n \times n$ matrices $K$ such that $$ s_A(K):=(-1)^{|A|}\det(K-I_A) \ge 0,\;\forall A \subseteq [\![n]\!], $$ where $I_A$ is masked version of the $n$-by-$n$ identity matrix defined by $(I_A)_{i,j}=\begin{cases}1,&\mbox{ if }i=j\text{ and }j \in A,\\{0},&\mbox{ otherwise.}\end{cases}$
Examples
One can show that every matrix of the form $K=VV^T(I_n + VV^T)^{-1}$, for some $V \in \mathbb R^{n \times k}$ with $k$ a positive integer, has the above property.
This generalization of an example proposed by user a user on MO. Any matrix of the form $K = \operatorname{diag}(t_1,\ldots,t_n) + X$ where $t_1,\ldots,t_n \in [0, 1)$ and $X$ is strictly upper triangular or strictly lower triangular (e.g $K=tI_n+X$, with $0 \le t < 1$). Indeed, $K-I_A=\operatorname{diag}(\tilde{t}_1,\ldots,\tilde{t}_n) + X$, where $$\tilde{t}_i = \begin{cases}t_i-1 < 0,&\mbox{ if }i \in A,\\ t_i \ge 0,&\mbox{ else.}\end{cases} $$ Thus, $\operatorname{sign}(\det(K-I_A)) = \operatorname{sign}(\Pi_{i=1}^n \tilde{t}_i)=\operatorname{sign}(\Pi_{i \in A}(t_i-1))\operatorname{sign}(\Pi_{i \not\in A}t_i)=(-1)^{|A|}$, from where it follows that $\text{sign}(s_A(K))=(-1)^{2|A|} = 1 > 0$ for all $A \subseteq [\![n]\!]$.
I was wondering whether one can get a large classes of matrices along similar lines (of simplicity and explicitness).
Observation
Also note that the set matrices $K$ verifying $$ s_A(K) > 0,\;\forall A \subseteq [\![n]\!] $$ is open in $\mathbb R^{n \times n}$.