When is the additive identity not the zero vector? My teacher cryptically mentioned today that the zero vector is not always the additive identity. When asked for clarification I was told "we'll get there".
He did confirm it is always 0 in matrices filled with real numbers, but I can't think of or find any matrix, whether complex or variable or whatever where anything else would work, or where the zero vector wouldn't work.
It might be half a joke to keep me interested, but I'll be a minkeys uncle if it didn't work! 
I don't know, any ideas?

That depends on what you mean by the zero vector. 
If you want, you can consider $\mathbb{R}_+$ (the set of positive real numbers) as a vector space, where you define $x\oplus y = xy$ and $\lambda x = x^\lambda$. Then the "additive identity" is actually $1$ (but should probably be called the zero vector in this strange context).
