The Material Point Method is an effective and efficient method for simulating scenarios with large deformations, such as impact or flowing. It describes the continuum as a set of particles moving through space, while their interactions are calculated on a Eulerian background grid. Therefore, instabilities due to mesh distortion are avoided and an efficient computation is maintained. Due to this discretization, however, the body often does not align with the grid. This complicates the proper imposition of boundary conditions, as the solution usually is computed using the grid nodes only. This contribution proposes a novel way to impose Dirichlet boundary conditions by means of the direct elimination method. It offers exact fulfillment of the constraints and reduces the number of degrees of freedom, positioning it favorably compared to other enforcement methods. The proposed methodology is applied to the implicit material point method using CPDI and WLS B-spline shape functions, offering improved stability and accuracy. The framework is validated and a possible application is demonstrated.