In this paper, we study the existence criteria for Ψ-bounded solutions of Sylvester matrix dynamical systems on time scales. The advantage of studying this system is it unifies continuous and discrete systems. First, we prove a necessary and sufficient condition for the existence of atleast one Ψ-bounded solution for Sylvester matrix dynamical systems on time scales, for every Lebesgue Ψ-deltaintegrable function F, on time scale T +. Further, we obtain a result relating to asymptotic behavior of Ψ-bounded solutions of this equation. The results are illustrated with suitable examples.