Thread: Proving matrix addition theorem

Hi, I'm in first year university and I'm new to proofs, so I was wondering if anyone here can help me out.

The questions ask to prove:

1) A + (B + C) = (A + B) + c

and

2) For each m x n matrix A, there is unique m x n matrix D such that

A + D = 0

Any hints and insights would be greatly appreciated!

Let's assume you're working over the real numbers.

Let , and . Define and .

By definition of the addition of matrices, we have , since real numbers are commutative. So, , and equivalently .

As for your second problem, letting , then set (i.e. each entry of matrix D is the additive inverse of the same entry in A). Then it follows that A + D is equal to the zero matrix, and you're done.

Futhermore, if you want to show that there is a UNIQUE matrix for 2), assume there are two different matrices which satisfy the condition, and work through all logical steps you can take - you'll get that they are equal to each other, and so your result is unique.

I think I understand now. Thanks for the help!