In the construction process while converting to a DFA, do not forget to account for the lambda transitions.

For example, the first step is to identify the initial state, say A

A $=\{0\}$ is not correct as the lambda transitions enable the starting to be $1.$ You can make a transition from $0$ to $1$ on lambda without scanning any symbol. Similarly, you can transition to $2,3,7.$

So the starting state of the DFA $=\{0,1,2,4,7\}$

The next step is to define transitions from A$,$ say A to B on symbol a$.$ Check that B $=\{1,2,3,4,6,7,8\}$

$($i$)$ Give a complete equivalent description of the DFA.

$($ii$)$ Convert the above to a minimal DFA.