A note of the representation of the Drazin inverse of 2x2 block matrix

[ A BCorollary 2.2 Let M =C DAB = 0 and CB = 0, thenM d =], where A ∈ C n×n and D ∈ C m×m . If[ A d + Y B(D d ) 2 ]X D d ,where X = X(D, C, A) is defined by (2) and Y = BXA d + BD d X.[ ] A BTheorem 2.5 Let M = , where A ∈ CC Dn×n and D ∈ C m×m . Ifone of the following conditions is satisfied:(1) A π B = B, ABD π = 0, CBD π = 0, D π C = C,(2) BD π = 0, A π B = B, D π C = C,(3) A π B = B, ABD π = 0, CBD π = 0, D d CA = 0 = 0, D d CB = 0,BD d C = 0,then M d can be represented as in (12).3 ExampleThe following example describes a 2 × 2 matrix M which does not satisfyconditions from Theorems 5.3 and Lemma 2.2, respectively of [9] and [11],whereas the conditions of Theorem 2.2 are met, which allows us to computeM D .[ ] A BExample. Consider a 2 × 2 block matrix M =, whereC D[ ][ ] [ ]1 01 11 0A = , B = D = and C = . Since BD ≠ 0 and0 00 0−1 0D is not nilpotent, the mentioned results from [9] and [11] fail to apply. Itis evident that BC = 0 and DC = 0, so we can apply Corollary 2.1, thusobtaining⎡⎤1 0 −1 −1M d = ⎢ 0 0 0 0⎥⎣ 1 0 −1 −1 ⎦ .−1 0 2 27