Is it permissible to secretly take money from the mother to spend on the daughter and her unmarried sisters for the purpose of studying and starting a project, given that the daughter knows the mother has a lot of money, and the mother denies this and refuses to spend on their education?

If the father is capable of providing for the family, then the sole responsibility for maintenance rests upon him, and it is not obligatory for the mother. However, if the father is in financial hardship, then maintenance becomes obligatory for the wealthy mother, and she cannot claim it back from the father even if he later becomes affluent.

If the father is capable but refuses to provide maintenance, then the mother must spend on her children, and she has the right to claim back from him what she has spent.

Maintenance is obligatory for daughters until they get married. It is permissible for one who has the right to maintenance to take from the provider's money secretly, within reasonable bounds, if the provider falls short in spending.

As for postgraduate studies, they are considered luxuries, and parents are not obligated to provide for them. It is not permissible to take money from the father or mother for this purpose without their permission. The primary role of the Muslim woman is to raise a believing generation and establish a Muslim household. It is highly probable that the mother's concealment of money is for the benefit of the children, thus one should have a good opinion of her and thank her.