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

In this paper, using an additive result for the Drazin inverese provedin [6],[we derive]formulae for the Drazin inverse of a 2 × 2 block matrixA BM = , under the conditions weaker than those in the papers [9],C D[11], [12] and [14].2 ResultsFirst, we present an additive result for the Drazin inverse proved in [6],which we will be useful in proving our main result.Theorem 2.1 Let P, Q ∈ C n×n be such that Q dconditions are satisfied= 0 and the followingP π Q = Q, P QP π = 0. (3)Then,∞∑(P + Q) d = P d + (P + Q) n Q(P d ) n+2 .n=0In the following theorem we derive a formula for the Drazin inverse ofblock-matrix M under some rather cumbersome and complicated conditionsbut the theorem itself will have a number of useful consequences which willinclude much simpler conditions.[ A BTheorem 2.2 Let M =C Dand], where A ∈ C n×n and D ∈ C m×m . IfD π C = C, BCA π = 0, DCA π = 0 (4)i D −1∑(A d ) n+1 BD n C = 0,n=0i∑D −1n=0i∑D −1n=0DC(A d ) n+1 BD n D π = 0,BC(A d ) n+1 BD n D π = 0,(5)3