It appears very clear, I think,that the identity referred to here should be the multiplicative identity. Every ring has an additive identity. In the case of these matrices if the following matrix is present then it has the multiplicative identity

[1 0]

[0 1].

Thus it clear that (a), (c), (d), & (f) cannot have that element.

On the other hand, both (b) & (e) can have that element.

But you say that the book says (c) has identity. Well if that is true the book may be talking about the additive identity! In that case each of the sets contains the element

[0 0]

[0 0].