Let E, F ∈ C n×n. If EF i E = 0 for all i ∈ N, we give the explicit representation of the Drazin inverse of the block complex matrix E I F 0. We thereby solve a wider kind of singular differential equations posed by Campbell [S.L. Campbell, The Drazin inverse and systems of second order linear differential equations, Linear & Multilinear Algebra, 14(1983), 195–198].