I think it is not wrong, but it is paraphrasing it rather than translating it. I think here we are supposed to recognize and translate the passive "by whom was it written" rather than paraphrasing it as an active "who wrote it". The meaning is the same, of course, but the grammatical construction differs
The hint translations for "da" are "return," "since," and "to". I suspect the correct translation is some clitic theyve not shown. So I'm not sure what "da" means in this context or what it's role in "da chi" is in the translation. These seriously need to be better. When it comes to verbs, especially, the root form should also been shown so we can deduce the various conjunctions and rules accross differing exercises (see, THAT'S how you learn, not getting it wrong and being told in the comments section).
“Da” is used to convey the agent by whom/which the action is performed when the VERB is in the PASSIVE voice.
Eg. Amleto è stato scritto da Shakespear = Hamlet was written by Shakespeare.
Il paese è stato distrutto dalla frano = The town was destroyed by the landslide.
Gobetti, D. (2006). Italian & prepositions. doi:10.1036/0071453938
After much searching, this is the only reference I could find regarding this peculiar use of “da”.