D: matrix(
 [1/E_x,-\nu_[xy]/E_x,-\nu_[xz]/E_x], 
 [-\nu_[yx]/E_y,1/E_y,-\nu_[yz]/E_y], 
 [-\nu_[zx]/E_z,-\nu_[zy]/E_z,1/E_z]
);

C:invert(D);

ratsimp(C);

C.D;

ratsimp(%o4);