i don’t know how to do it with matricies, but if i understand the problem correctly, you could just add up all the distances from the target point to each vertex, then divide each distance by that total, to get normalized weights for each vert.
then you take the change in position for each vertex, from the old tetrahedron to the new tetrahedron, and multiply each vector by its respective weight, total the resulting vectors, then add it to the target points previous location. that should skin the point to the deformed mesh.