I'm taking a linear algebra class at my university and recently we have been talking about the multiplicative inverse of matrices and how some matrices do not have multiplicative inverses (i.e. matrices that aren't square). This made me wonder whether vectors have inverses as well, specifically additive inverses. Isn't the additive inverse of a vector just the negative of that same vector, and isn't the sum of a vector and its additive inverse just the zero vector? In that case, is the statement below true in all cases? And does the zero vector have an inverse?
"Every vector must have an additive inverse, the sum of these two vectors being the zero vector."