In　the　paper,　we　propose　a　deterministic　approximation　algorithm　for　maximizing　a　generalized　monotone　submodular　function　subject　to　a　matroid　constraint.　The　function　is　generalized　through　a　curvature　parameter,　and　essentially　reduces　to　a　submodular　function　when.　Our　algorithm　employs　the　deterministic　approximation　devised　by　Buchbinder　et　al.[3]　for　the　case　of　the　problem　as　a　building　block,　and　eventually　attains　an　approximation　ratio　of　for　the　curvature　parameter　and　for　a　calibrating　parameter　that　is　any.　For,　the　ratio　attains　0.5008　by　setting,　coinciding　with　the　renowned　performance　guarantee　of　the　problem.　Moreover,　when　the　submodular　set　function　degenerates　to　a　linear　function,　our　generalized　algorithm　always　produces　optimum　solutions　and　thus　achieves　an　approximation　ratio　1.　©　2020,　Springer　Nature　Switzerland　AG.