$S=QQ,Q$ is the set of rational numbers, $*$ is a binary operator on $,ay+b>.$

$(1)$ Determine whether or not the operation $*$ on $S$ is commutative, associative, and idempotent?

$(2)$ Determine whether or not the operation $*$ on $S$ has identity and zero element? If it has, please list them, and further find the inverse element of every invertible element in $S.$

Let $S=\{a,b\},$ there are $4$ operators on $S$ : ${?}^{**},,.,$ and $,$ as shown in the following table.

$\backslash$table$[[+,a,b$