Some structural properties of a general Petri net (PN) are considered. The paper endeavors to improve the link between PNs, the theory of matrix analysis, and linear inequalities. Necessary and/or sufficient conditions for consistency, conservativeness, boundedness, and repetitiveness are given in terms of certain determinants of the incidence matrix of the net. Besides, when the incidence matrix is square, theorems are derived in terms of the modified-incidence-matrix eigenvalues and, thus, inherently reduce the state-space explosion problem. Examples are worked out to illustrate the results.