(i) D is Diagonalizable
This one i believe to be fairly straightforward, if D is diagonalizable then we can allow $D^t = I$ (where I is the identity) and therefore D id diagonalizable and therefore A=N+D is straightforward.
(ii)N is Nilpotent
So when there exists an r such that $N^r=0$
(iii)DN = ND
Each one seems rather trivial, e.g for (ii) D could be any square matrix of the same size as N and it be a Square matrix A and for (iii) you just prove commutativity for matrices?
I believe I'm missing something quite important here but can't figure out what