  Using the $det^{S^2}$ map from [5], we introduce the notion of $S^2$-rank of
a matrix of type $d\times \frac{s(s-1)}{2}$. As an application, we show that
the conditional probability matrix associated to two random variables has the
$S^2$-rank equal to $1$. Under suitable conditions we prove that the converse
of this result also holds.
