  Principal matrices of a numerical semigroup of embedding dimension n are
special types of $n \times n$ matrices over integers of rank $\leq n - 1$. We
show that such matrices and even the pseudo principal matrices of size n must
have rank $\geq \frac{n}{2}$ regardless of the embedding dimension. We give
structure theorems for pseudo principal matrices for which at least one $n - 1
\times n - 1$ principal minor vanish and thereby characterize the semigroups in
embedding dimensions $4$ and $5$ in terms of their principal matrices. When the
pseudo principal matrix is of rank $n - 1$, we give a sufficient condition for
it to be principal.
