Here is a quick way to create a symmetric matrix in Mathematica. The code below creates a generic 6×6 matrix (line 1), then mirrors the above-diagonal elements to their below-diagonal counterparts (substitution in lines 3-5).
am = Array[Subscript[a,##] &, {6,6}];
am = am /.
Flatten[Table[
Subscript[a,i,j] -> Subscript[a,j,i],
{i,2,6}, {j,1,i-1}]]
Create a symmetric matrix in Mathematica