The most common method to design tuned dynamic vibration absorbers is still that of Den Hartog, based on the principle of invariant points. However this method is optimal only when attaching the absorber to a single-degree-of-freedom undamped main system. In the present paper an extension of the classical Den Hartog approach to a multi-degree-of-freedom undamped main system is presented. The Sherman-Morrison matrix inversion theorem is used to obtain an expression that leads to invariant points for a multi-degree-of-freedom undamped main system. Using this expression, an analytical solution for the optimal damper value of the absorber is derived. Also, the effect of location of the absorber in the multi-degree-of-freedom system and the effect of the absorber on neighboring modes are discussed.