The reverse order law for the Drazin inverses of multiple matrix products (Q1601636)

It is well known that the Drazin inverse has been widely applied to the theory of finite Markov chains and singular differential and difference equations. In a classic paper \textit{T. N. E. Greville} [SIAM Rev. 8, 518-521 (1966; Zbl 0143.26303)] gave necessary and sufficient conditions of the reverse order law for the Moore-Penrose inverse \((AB)^{+}= B^{+}A^{+}\) to hold for two complex matrices \(A\) and \(B\). In general, the reverse order law does not hold for the Drazin inverse, that is \((AB)^{D} \neq B^{D}A^{D}\). Drazin proved that \((AB)^{D} = B^{D}A^{D}\) holds under the condition \(AB = BA\). In the paper under review the author gives necessary and sufficient conditions for the \(n\) term reverse order law \((A_1A_2\cdots A_n)^{D} = A_n^DA_{n-1}^D \cdots A_2^D A_1^D\) in terms of some rank equality.