"My wife was in Beijing last week."
The way I was taught is that, yes, 了 can be used to make a verb past tense but ONLY if it is an action verb like walk, see, eat etc. 在 can be a verb but it isn't an action verb.
I've always seen 了 after the verb. Maybe you thought about the adverbs that do come before the verb.
I know 了isn't supposed to be a indicator of past tense, but since the wife *was in Beijing last week wouldn't that mean the action was completed?
I could be wrong, but I'm fairly sure you can drop the 的 when it's someone close such as a wife (i.e. 我妻子上个星期在北京)
Isn`t 上个星期我妻子去了北京. also correct? And shouldn't this be preferred over 在 since the context implies that the action is finished?
Your sentence would probably be disqualified because you said 我妻子 which is quite common but seems to be consistently rejected by Duolingo. Recommend that you always add the 的 while on Duolingo: 我的妻子.
The main point of your comment, though, is about 去了 vs 在. 去了indicates that your wife travelled to Beijing last week, that is to say, she completed her trip from somewhere to Beijing. 在 simply indicates that she was there; she could have been there for the whole week, she might have been there for a year, she might still be there now, or she might have been there for only a day.
I don't think the English implies anything about where your wife was prior to, or since, last week.
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There is no indication in the English sentence that your wife is no longer in Beijing, from the perspective of last week, the action is continuing. Anyway, using 在了 is pretty uncommon, usually if you wanted to indicate arrival somewhere you would say 來了 or 去了 or 到了.