In general, if the subject of the sentence is a personal pronoun, it has to agree with the gender of the object. But this can also depend on the context.
For example, in this sentence, it is not clear if the child is a boy or a girl, so all genders are acceptable in the subject. So you could say:
- Αυτός είναι το παιδί μας = He is our child
- Αυτή είναι το παιδί μας = She is our child
- Αυτό είναι το παιδί μας = It/This is our child
I hope that this helps. :-)
This is not very correct, I'm afraid.
In the above sentences, "το παιδί μας" is not an object, it is rather the predicate ("κατηγορούμενο"), which means that the verb gives information about a category (hence "κατηγορούμενο") in which the subject belongs. In this case, the category is "our children" and it contains only one member, because of the definite article "το" (if we had more children, we would say "Αυτός/αυτή/αυτό είναι παιδί μας"). There are specific verbs that are followed by a predicate (e.g. be, become).
When the predicate is an adjective, it has to agree with the gender and number of the subject:
Αυτός είναι καλός = He is good
Η Ελένη είναι καλή = Helen is good
Αυτοί είναι καλοί = They (masc.) are good, etc.
However, when the predicate is a noun, the subject may be in any gender or number, be it a personal pronoun or not. The above examples with "το παιδί μας" demonstrate this. Some further examples:
Αυτή είναι καμήλα (they agree) but:
Αυτό το ζώο είναι καμήλα
Αυτές είναι ομάδα = They (fem.) are a team (sing. fem.)
Η ζωή είναι ένα ταξίδι = Life (fem.) is a travel (neut.)
Αυτό είναι μικροπρέπεια = This is pettiness (fem.), etc.
When a verd is followed by an object (which means that the verb shows that the subject causes some kind of change to the object), the gender and number of the subject are unrelated to those of the object, as can be shown by sentences already seen in this course:
Αυτή τρώει ένα μήλο
Το παιδί τρώει ζυμαρικά
Ο αετός πίνει νερό
Τα παιδιά διαβάζουν εφημερίδα, etc.
Sorry for the grammar overdose! :p