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In this study, we derive the Drazin inverse(A+εB)D{\left(A+\varepsilon B)}^{D}of the complex matrixA+εBA+\varepsilon BwithInd(A+εB)&gt;1{\rm{Ind}}\left(A+\varepsilon B)\gt 1andInd(A)=k{\rm{Ind}}\left(A)=kand the group inverse(A+εB)#{\left(A+\varepsilon B)}^{\#}of the complex matrixA+εBA+\varepsilon BwithInd(A+εB)=1{\rm{Ind}}\left(A+\varepsilon B)=1andInd(A)=k{\rm{Ind}}\left(A)=kwhenεB\varepsilon Bis viewed as the perturbation ofAA. If the dual Drazin inverse (DDGI)A^DDGI{\widehat{A}}^{{\rm{DDGI}}}ofA^\widehat{A}is considered as a notation. We calculate(A+εB)D−A^DDGI{\left(A+\varepsilon B)}^{D}-{\widehat{A}}^{{\rm{DDGI}}}and(A+εB)#−A^DDGI{\left(A+\varepsilon B)}^{\#}-{\widehat{A}}^{{\rm{DDGI}}}and obtain‖(A+εB)D−A^DDGI‖P∈O(ε2)\Vert {\left(A+\varepsilon B)}^{D}-{\widehat{A}}^{{\rm{DDGI}}}{\Vert }_{P}\in {\mathcal{O}}\left({\varepsilon }^{2})and‖(A+εB)#−A^DDGI‖P∈O(ε2)\Vert {\left(A+\varepsilon B)}^{\#}-{\widehat{A}}^{{\rm{DDGI}}}{\Vert }_{P}\in {\mathcal{O}}\left({\varepsilon }^{2}). Meanwhile, we give some examples to verify these conclusions.

The perturbation of Drazin inverse and dual Drazin inverse