Prove $D_0^TW^{-1}D_0$ is nonsingular.
Where $W$ is a $l$x$l$ nonsingular(of course) symmetric, finite and positive definite matrix. And $D_0$ is a full rank $l$x$k$ matrix with rank $k$
Use Slyvester's rank inequality I can get $k\le rank(D_0^TW^{-1})\le k$.
Use the result above $rank(D_0^TW^{-1})=k$ and apply Slyverster's theorem again I can only get $2k-l \le rank(D_0^TW^{-1})\le k$.
I think I need to use more information of $W$ but I don't know how.
Source: I got this problem from a econometrics textbook and there are some UWLLN(uniform weak law of large numbers) of the sample which I think is irrelevant to the proof, so I didn't list it here. So if you think this question can't be proved by the assumptions above, please tell me so that I can add more details about the assumptions.
The matrix $D^T W^{-1}D$ is symmetric. We test for positive definiteness as follows: take $x\ne0$. Then $$ x^TD^T W^{-1}Dx = (Dx)^TW^{-1}Dx. $$ Since $W$ is positive definite, $W^{-1}$ is positive definite as well. This shows $$ x^TD^T W^{-1}Dx \ge0 $$ and $$ x^TD^T W^{-1}Dx =0 \Leftrightarrow Dx=0. $$ The matrix $D$ has full column rank, hence $Dx=0$ is equivalent to $x=0$.
Hence $D^T W^{-1}D$ is symmetric positive definite, thus invertible.