$(1)$ What is number of rank $n$ $m\times n$ $0-1$ matrices when $m=n$ and $m>n$? Is there solutions closed forms?
$(2)$ Over $\Bbb F_2$, does full rank of an $n\times n$ matrix imply determinant is non-zero and vice versa?
2026-03-28 06:59:07.1774681147
On full rank matrix properties
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For $m=n$ you might look at OEIS sequence A055165 and references there, in particular the Živković article. Bottom line: there seems to be no known closed form, but there are conjectured asymptotics.
EDIT: For question (2): Over any field, full rank of an $n \times n$ matrix is equivalent to the determinant (evaluated in that field) being nonzero.