A note on the Moore-Penrose inverse of block matrices

Abstract

Motivated by the representation for the Moore-Penrose inverse of the block matrix over a $\ast$-regular ring presented in [R.E. Hartwig and P. Patr\'{i}cio, When does the Moore-Penrose inverse flip?
Operators and Matrices, 6(1):181-192, 2012], we show that
the formula of the Moore-Penrose inverse is the same as the expression given by [Nieves Castro-Gonz\'{a}lez, Jianlong Chen and Long Wang, Further results on generalized inverses in rings with involution, Elect. J. Linear Algebra, 30:118-134, 2015].