A note on the representations for the Drazin inverse of \(2 \times 2\) block matrices (Q880041)

Given a matrix \(M = \left[ {\begin{matrix} A & B \\ C & D \\ \end{matrix} } \right]\), where \(A \in {\mathbb C}^{m \times m} \) and \(D \in {\mathbb C}^{n \times n}\), a new formula for the Drazin inverse \(M^D\) is derived under the assumptions that: \(A(I - AA^D)B = 0, C(I - AA^D)B = 0, BCAA^D = 0, DCAA^D = 0\).