i have no expertise on this topic but if its a general case then i wouldn't expect that you can put any specific values in D. You can leave them as unknowns (eg, etc).
anyway. For the first part of your problem:
let which is a scalar
I think you need to show that has an inverse, given that DD' has an inverse. I have no idea how to do that but since this is now a linear algebra problem perhaps some other forum user can finish.
For the second part, it seems clear from general reasoning that R* = R - 1 because in the revised market as every other asset has had its rate of return reduced by 1 in every state.