Drazin Inverse of a $$2 \times 2$$ Block Matrix

The problem of finding representations of the Drazin inverse of a \(2\times 2\) block matrix is of great significance primarily due to its applications in solving systems of linear differential equations, linear difference equations [1, 2, 3, 4] and perturbation theory of the Drazin inverse [5, 6, 7].