In this paper, we propose the inequalities which are satisfied by the determinant of the solutions to the systems of matrix equations \[ \begin{cases} AXB^T+BX^TA^T=C, DXE^T+EX^TD^T=F,\quad(I) \end{cases}\qquad \begin{cases} AXB^T+BXA^T=C, DXE^T+EXD^T=F.\quad (II) \end{cases} \] By applying the proposed inequalities, lower bounds for the product of the eigenvalues of the solutions to the systems of matrix equations (I) and (II) are established.