In　this　paper,　we　study　the　generalized　submodular　maximization　problem　with　a　non-negative　monotone　submodular　set　function　as　the　objective　function　and　subject　to　a　matroid　constraint.　The　problem　is　generalized　through　the　curvature　parameter　α∈[0,1]　which　measures　how　far　a　set　function　deviates　from　linearity　to　submodularity.　We　propose　a　deterministic　approximation　algorithm　which　uses　the　approximation　algorithm　proposed　by　Buchbinder　et　al.　[2]　as　a　building　block　and　inherits　the　approximation　guarantee　for　α=1.　For　general　value　of　the　curvature　parameter　α∈[0,1],　we　present　an　approximation　algorithm　with　a　factor　of　[Formula　presented],　where　y∈[0,1]　is　a　predefined　parameter　for　tuning　the　ratio.　In　particular,　when　α=1　we　obtain　a　ratio　0.5008　when　setting　y=0.9,　coinciding　with　the　renowned　state-of-art　approximate　ratio;　when　α=0　that　the　object　is　a　linear　function,　the　approximation　factor　equals　one　and　our　algorithm　is　indeed　an　exact　algorithm　that　always　produces　optimum　solutions.　©　2021　Elsevier　B.V.