We study the sum degrees of freedom (DoFs) of the K-user rank-deficient interference channel with feedback. For the two-user case, we characterize the sum DoF by developing an achievable scheme and deriving a matching upper bound. For the three-user case, we develop a new achievable scheme which employs interference alignment to efficiently utilize the dimension of the received signal space. In addition, we derive an upper bound for the general K-user case and show the tightness of the bound when the number of antennas at each node is sufficiently large. As a consequence of these results, we show that feedback can increase the DoF when the number of antennas at each node is large enough as compared with the ranks of channel matrices. This finding is in contrast to the full-rank interference channel where feedback provides no DoF gain. The gain comes from using feedback to provide alternative signal paths, thereby effectively increasing the ranks of desired channel matrices.