On page 209 of Gelman and Hill book, the authors suggest that
Having created and checked appropriateness of the matches by examining balance, we fit a regression model just on the matched data including all the predictors considered so far, along with an indicator to estimate the treatment effect
Why do I still need to include predictors? After matching, all of these predictors are balanced across the control and treatment group, and thus uncorrelated with the treatment variable. Therefore, the coefficient estimate should be no different with and without the predictors.