The consistency and the exact solutions to a system of matrix equations

We provide two techniques, which establish criteria for the consistency of a system of matrix equations (see (1.1)). The system encompassesmatrix systems that were not studied before. We present the solutions set of a consistent system (1.1), using each technique. We investigate the link between the two techniques. We study the number of solutions that a system (1.1) could have, and establish a necessary and sufficient condition for a consistent system (1.1) to have a unique solution. If A(i), B-i and C-i, i = 1, ..., s, are all the zero matrices in (1.1) and the system is consistent, we provide bounds for the rank and inertia of any Hermitian solution of the system.