Number of Positive Definite Binary Matrices

Question

How may positive definite matrices (over finite field- $F_p$) are possible? What is the criterion in getting those?

Answer 1

If the problem is counting the binary matrices $M$ (all entries zero or one) that are positive definite symmetric as real matrices, the answer seems to be only the $n \times n$ identity matrix for each $n$.

Certainly the diagonal must consist of ones, else $e_i^T M e_i$ would be zero for some standard basis vector $e_i$.

If some off-diagonal matrix entry is $1$, then we have a singular $2\times 2$ principal minor, thus disproving the positive definiteness of $M$.

If the problem involves counting matrices over a finite field, there is no "standard definition that works" of positive definite matrix. Indeed there is no notion of positive that works in a finite field, since for characteristic $p$ one gets:

$$ 1 + 1 + \ldots + 1 = 0 $$

for $p$ copies of summand $1$, so while $1 = 1^2$ ought to be "positive", it isn't (by the definition used for ordered fields).