This can also mean: "Their women." That's what I typed and it counted it as wrong. The same word is used for "woman" and "wife" in Greek. Though I understand that when you say "η γυναικα ΜΟΥ" it makes more sense for it to mean "wife" instead of "woman," when ownership is shown like that, I still do not believe the answer I gave should be marked incorrect.
We have "women/wives/ladies" for every use of "γυναίκα" and it is included here as an alternative. Did it say you were wrong or "another correct sentence" or "almost correct"? These do not mean wrong. If you hover over γηναίκα above you'll find at the bottom "women". You have to scroll.
If you typed it exactly like so, I'm not surprised that it was marked incorrect. :P
Please make sure that there are no typos in your answers before the final check ^.^
The meaning is very similar however, "δικές τους" "their own" shows more emphasis. We could have presented "Οι γυναίκες τους." but we are trying to give you practice is a variety of forms. So, since it says, "...δικές τους..." you need "their own".
That's ok because they both sound exactly the same. As for the use you can find a full list of the uses of the articles including Η and Οι if you go back to Basic 1 and read the Tips & notes (found just under the word list on the Basic 1 Homepage).
Which is quite logical. Η is feminine singular, Οι is feminine plural. both δικές and γνώμες are feminine plural, so the article would have to match the number and gender. Οι δικές τους γνώμες is grammatically correct. ^.^
If you had read the Tips & notes I suggested above you would have seen as Dimitra explains that "Η" is singular and "Οι" is plural.
I looked for that, but they don't have it on the app. I don't want you to think I'm ignoring your suggestion, I just don't do Duolingo on computer.
Correct in the plural and for a fuller view go back to Basic 1 and check out the Tips & notes which you'll find on the homepage. There is a good explanation and of other important issues. In fact, you should always read the Tips & notes you'll find lots of useful information.