Let A be a 3x3 matrix and M be the matrix representation of my inner product.
I have to check if the map associated to A is self-adjoint.
This means: $$\langle Ax,y \rangle = \langle x,Ay\rangle$$.
Now would it be enough to prove that M (matrix reprensentation of my inner product) is symmetric? Does then follow that it is self-adjoint.