Determinant Question (Proof)

Question

Let $C$ and $D$ be $n \times n$ matrices where n is odd such that $CD = -DC$. Show that either $C$ or $D$ has no inverse.

I have no idea how to go about doing this problem. Any help would be appreciated.

Answer 1

Hint $\det(AB)=\det(A)\det(B)$; in particular $$ \det(C)\det(D)=-\det(C)\det(D) $$ What can you argue from this?

Answer 2

$$CD=−DC\\det(CD=−DC)\\det(CD)=det(−DC)\\det(CD)=(−1)^n det(DC)\\det(C)det(D)=(-1)^n det(C)det(D)\\det(C)det(D)-(-1)^n det(C)det(D)=0 \\det(C)det(D)(1-(-1)^n)=0\\now\\n\\is\\odd\\so\\det(C)det(D)(1-(-1))=0\\det(C)det(D)(2)=0\\so\\det(C)det(D)=0\\det(C)=0\\or\\det(D)=0 $$