In this paper, we investigate the recent paper of Shakoor et al. [A. Shakoor, I. Ali, S. Wali, A. Rehman, Some formulas on Drazin inverse for sum two matrices and block matrices, Bull. Iran. Math. Soc. 48 (2022) 351-366]. Here prove that main, additive result from mentioned is actually a corollary one known result. Furthermore, give new representations anti-triangular matrix, which generalize s...

For a matrix over a complex commutative unital Banach algebra, necessary and suucient conditions are given for the existence of its group inverse, and more generally, its Drazin inverses. The conditions are easy to check and explicit formulas for the inverses are provided. Some properties of the inverses and an application to operator theory are discussed. This note is a continuation of an earl...

We apply single and double tree-like representations of Markov jump processes on $\mathbb{Z}_N$ for obtaining their nonequilibrium heat capacity taking the diffusion limit $N\uparrow \infty$. The main tool is a graphical representation Drazin-inverse backward generator. In that way, combination algebraic graph-theoretical approaches enables an exact computation important quantity (excess heat) ...