On the existence of a 'strong' type of of invertible elements of $\mathcal{M}_n(\mathbb{K})$

Question

For a given field $\mathbb{K}$ and a positive integer $n$, is it possible to find a non-singular matrix in $\mathcal{M}_n(\mathbb{K})$ that remains non-singular no matter how its entries are permuted? What if the field is $\mathbb{R}$? or algebraically closed (e.g. $\mathbb{C})$? What if the entries are restricted to an interval of the real line (e.g. $[1,2]$)? What if the field is finite (e.g. $\mathbb{Z}_p$ for prime $p$)?

Answer 1

EDIT: This is an anwer to the original question where the field is $\mathbb{R}$.

Sure. You have a finite number ($n^2!$) of multilinear polynomials in the matrix entries that must all be nonzero. It is immediate that the set of matrices satisfying your property is open, and not hard to argue that almost every matrix will satisfy your property.

Your question is a lot more interesting over, say, a finite field.