I need to prove that for each $\displaystyle m \times n$ matrix A, there exists a unique $\displaystyle m \times n$ matrix D such that A+D=O. Where O is the $\displaystyle m \times n$ zero matrix.

I have a textbook that states the proof simply as:

$\displaystyle \textrm{For } A=[a_{ij}] \textrm{ Let } D=[-a_{ij}]. $

I'm not sure how to construct a a better proof for this result. Could anyone offer a starting point?