*(English)*Zbl 0998.15010

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 *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 ${\left(AB\right)}^{+}={B}^{+}{A}^{+}$ to hold for two complex matrices $A$ and $B$. In general, the reverse order law does not hold for the Drazin inverse, that is ${\left(AB\right)}^{D}\ne {B}^{D}{A}^{D}$. Drazin proved that ${\left(AB\right)}^{D}={B}^{D}{A}^{D}$ 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 ${({A}_{1}{A}_{2}\cdots {A}_{n})}^{D}={A}_{n}^{D}{A}_{n-1}^{D}\cdots {A}_{2}^{D}{A}_{1}^{D}$ in terms of some rank equality.