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

. A is an orthogonal matrix if

(since

and

are symmetric

and

commute). And must have an inverse

.

Let

denote the rows of

; then

are the columns of

.

Let

By matrix multipication:

Therefore

rows of A form an orthonormal set.

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