For $R$ to be an equivalence relation on $A = \{1, 2, 3, 4\}$, it must be reflexive, symmetric and transitive.

1. **Reflexive:** $R$ must contain $(1, 1), (2, 2), (3, 3), (4, 4)$. Since $(3,3)$ is already in $R$, we need to add $(1, 1), (2, 2), (4, 4)$.

2. **Symmetric:** $R$ must contain $(2, 1)$ and $(3, 2)$ because it contains $(1, 2)$ and $(2, 3)$.

3. **Transitive:** Since $(1, 2)$ and $(2, 3)$ are in $R$, $(1, 3)$ must also be in $R$. And since we added $(3,2)$ now we must add $(1,2)$. Which already exists.

So, the minimum elements to be added are:
$(1, 1), (2, 2), (4, 4), (2, 1), (3, 2), (1, 3)$.

Therefore, the minimum number of elements to be added is $7$.