Convolutional codes form an important class of codes that have memory. One natural way to study these codes is by means of input state output representations. In this paper we study the minimum (Hamming) weight among codewords produced by input sequences of weight two. In this paper, we consider rate $1/n$ and use the linear system setting called $(A,B,C,D)$ input-state-space representations of convolutional codes for our analysis. Previous results on this area were recently derived assuming that the matrix $A$, in the input-state-output representation, is nonsingular. This work completes this thread of research by treating the nontrivial case in which $A$ is singular. Codewords generated by weight-2 inputs are relevant to determine the effective free distance of Turbo codes.