Linear Algebra Proof - Prove that when A+D=0, D is unique.

**Fourier** · September 3rd 2007, 10:14 AM

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

I have a textbook that states the proof simply as:

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

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

**red_dog** · September 3rd 2007, 11:22 AM

Let $A=(a_{ij}), \ D=(x_{ij})$
Then $A+D=(a_{ij}+x_{ij})$ .
$A+D=O_{mn}\Rightarrow a_{ij}+x_{ij}=0\Rightarrow x_{ij}=-a_{ij}\Rightarrow D=(-a_{ij})$

Suppose $D'$ is another matrix such as $A+D'=D'+A=O_{mn}$ .
Then $D'=D'+O_{mn}=D'+(A+D)=(D'+A)+D=O_{mn}+D=D$

**Fourier** · September 4th 2007, 04:28 PM

Thanks for giving the alternate proof.

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