Our former study Tóbiás and Tasi (J Math Chem 54:85, 2016) is continued, where a simple algebraic solution was given to the kinetic problem of triangle, quadrangle and pentangle reactions. In the present work, after defining chemical reaction networks and their connectedness, first-order chemical reaction networks (FCRNs) are studied on the basis of the results achieved by Chellaboina et al. (Control Syst 29:60, 2009). First, it is proved that an FCRN is disconnected iff its coefficient matrix is block diagonalizable. Furthermore, mass incompatibility is used to interpret the reducibility of subconservative networks. For conservative FCRNs, the so-called marker network is introduced, which is linearly conjugate to the original one, to describe the zero eigenvalue associated to the coefficient matrix of an FCRN. Instead of using graph-theoretical concepts, simple algebraic tools are applied to present and solve these problems. As an illustration, an industrially important ten-component (formal) FCRN is presented which has algebraically exact solution.