Representation of the Drazin inverse for special block matrix

For A∈Cn×nA∈Cn×n, the matrix X∈Cn×nX∈Cn×n is said to be the Drazin inverse of A  , if it holds that XAX=X,AX=XA,ArXA=Ar, where r   is the smallest nonnegative integer such that rank(Ar)=rank(Ar+1)rank(Ar)=rank(Ar+1). In this paper, we give the representation for the Drazin inverse of the block matrix AAB0(A,B∈Cn×n,A2=A).