I guess A^T = A would be true for every inner product. As by its defn, it should satisfy <u,v> = <v,u>. Therefore, u1v2 and u2v1 should have the same coefficient, and hence the upper right and lower left entry should have the same value, so it would always be symmetric.
I think this is because that inner product should be symmetric by definition.
bacause only when A=AT, <u,v> can be equal to <v,u>
Since <u,v> = <v,u>. If it is not symmetric, it doesn't not obey the definition.
Yes, because <u,v> = <v,u>.