Since relation needs to be reflexive the ordered pairs $(1,1),(2,2),(3,3)$ need to be there and $(1,2)$ is also to be included.

Let's call $R_0=\{(1,1),(2,2),(3,3),(1,2)\}$ the base relation.

$\because A \times A$ contain $3 \times 3=9$ ordered pairs, remaining 5 ordered are

$$2,1),(1,3),(3,1),(2,3),(3,2)$$

We have to add at most two ordered pairs to $R_0$ such that resulting relation is reflexive, transitive but not symmetric.

Following are the only possibilities.

$$R=R_0 U\{(1,3)\}$$

OR $R_0 U\{(3,2)\}$

OR $R_0 U\{(1,3),(3,1)\}$

OR $R_0 U\{(1,3),(3,2)\}$

OR $R_0 U\{(3,1),(3,2)\}$