consider a 5*5 square matrix, where entries in each cell represent the number of ways we can fill it. Since only 0 or 1 can be inserted therefore 2 ways.
$\bigl(\begin{smallmatrix} 2& 1 & 1& 1 &1\\ 2& 2 &1 &1 &1 \\ 2&2 &2 &1 & 1\\ 2&2 & 2 &2 &1 \\ 2&2 &2 & 2 & 2 \end{smallmatrix}\bigr)$
Now diagonal entries can be filled by any of 0 or 1 , so 2 ways for diagonal elements
but for non-diagonal we need to fix the elements in either upper/lower triangle of matrix, Because its a symmetric matrix hence it will be mirror image about diagonal.
for e.g. if A[2,1]= 1, then A[1,2]=1 OR if A[2,1]= 0, then A[1,2]=0
fixing lower triangle: number of ways to fill A[2,1] is 2 , so number of ways to fill A[1,2] is 1
otherwise fixing upper triangle: number of ways to fill A[1,2] is 2, so number of ways to fill A[2,1] is 1
number of n*n symmetric = $2^{n}*2^{n-1}*2^{n-1}$............$2^{1}$
=$2^{\frac{n(n+1)}{2}}$