This technical note deals with a modified algebraic Riccati equation (MARE) and its corresponding inequality and difference equation, which arise in modified optimal control and filtering problems and are introduced into the cooperative control problems recently. The stabilizing property of the solution to MARE is presented. Then, the uniqueness is proved for the almost stabilizing and positive semi-definite solution. Next, the parameter dependence of MARE is analyzed. An obtained parameter dependence result is finally applied to the study of semi-global synchronization of leader-following networks with discrete-time linear dynamics subject to actuator saturation.