1. The problem statement, all variables and given/known data Let V= set of 2x2 matrices with the normal addition, but where multiplication is defined as: β#A=β(A^T) where A^T is the transpose of A. 2. Relevant equations The axiom about 1#A=A 3. The attempt at a solution I think that because you can show that not ALL matrices satisfy A=A^T, you can't have a vector space since the multiplication by 1 doesn't hold up. But then I'm wondering whether I'm assuming that the multiplicative identity should be the "normal" 1 (ie that 1 is just the scalar 1 in a normal R^n vector space). How do you prove a multiplicative identity absolutely does NOT exist?