Solution

The language $L={a,b}^*$ means all possible strings over $a$ and $b$, including null string $\epsilon$.

So, an automaton recognizes $L={a,b}^*$ if every string made from $a$ and $b$ is accepted.

In A, $q_0$ is the initial and final state, and both $a$ and $b$ return to $q_0$.

So, A accepts all strings over ${a,b}$.

In B and C, $q_0$ and $q_1$ are final states, so all possible moves remain in final states.

So, B and C also accept all strings over ${a,b}$.

In D, $q_0$ is final and from $q_0$, both $a$ and $b$ return to $q_0$. Other states do not affect acceptance because starting from $q_0$, every input stays at $q_0$.

So, D also accepts all strings over ${a,b}$.

Therefore, A, B, C and D recognize $L={a,b}^*$.