In　this　paper,　we　consider　the　problem　of　maximizing　a　non-submodular　set　function　subject　to　a　matroid　constraint　with　the　continuous　generic　submodularity　ratio　γ.　It　quantifies　how　close　a　monotone　function　is　to　being　submodular.　As　our　main　contribution,　we　propose　a　(1-e-γ2-O(Ε))　-approximation　algorithm　when　the　submodularity　ratio　is　sufficiently　large.　Our　work　also　can　be　seen　as　the　first　extension　of　the　adaptive　sequencing　technique　in　non-submodular　case.　©　2020,　Springer　Nature　Switzerland　AG.