The idea is right, and perhaps your thinking is perfectly sound. Let me mention a few details anyway.
Remember that for a symmetric relation , it holds by definition that if , then . So to prove the direction that you label as "i", you assume that is a symmetric matrix, and to prove that the relation is then symmetric, you need to assume that for some . From this assumption, you need to prove that . For this purpose, your argument works fine: Since , we have . Since the matrix is symmetric, we have also , which proves that .
Similarly for the second part. Does your proof show (assuming is symmetric) that if , then ?