So, I tried to prove this question in two parts (A) if A is a symmetric matrix then it is orthogonal (B) since A is orthogonal, its row vectors must form an arthonormal set. But I don't think it is possible to prove part (A).

Anyway here's my proof so far:

If A is symmetric then $\displaystyle A=A^T$. A is an orthogonal matrix if $\displaystyle AA^T=A^TA=I$ (since $\displaystyle A$ and $\displaystyle A^T$ are symmetric $\displaystyle A$ and $\displaystyle A^T$ commute). And must have an inverse $\displaystyle A^{-1}=A^T$.

Let $\displaystyle R_1,R_2,...R_n$ denote the rows of $\displaystyle A$; then $\displaystyle R^T_1,R^T_2,...R^T_n$ are the columns of $\displaystyle A^T$.

Let $\displaystyle AA^T=C_{ij}$

By matrix multipication: $\displaystyle C_{ij}=R^T_iR^T_j=R_i.R_j$

Therefore $\displaystyle AA^T=I \iff R_i . R_j = \delta_{ij} \iff$ rows of A form an orthonormal set.

My proof is proof is probably not correct. Can anyone show me how to prove this?