Proving Orthogonality of Rigid Body Modes

I'm working on a MSc thesis involving both rigid body (zero energy) & deformational (elastic) modes and their orthogonality wrt mass matrix. I want to prove that elastic modes are mutually orthogonal (done !), that elastic & rigid body modes are so (sort of done !) & that rigid body modes also are so (unsure how ?). Can anyone offer any advice ?

Thanks, John.