$CD+C+D=0$, show that $CD=DC$

Question

So say we have two ($n \times n$)-matrices $C$ and $D$ and also that $CD+C+D=0$

Prove that $CD=DC$.

I'm kind of curious about this one and don't know where to start.

Answer 1

Hint: Add $I$ to both sides of the original equation, and factor the left-hand side to get $$ (C+I)(D+I)=I $$ Now remember that inverses are always two-sided.