Is it Islamically permissible for the brother of the deceased wife to hand over a sum of money to a merchant for investment, and for the profits to be distributed monthly to the poor as a continuous charity on behalf of the deceased wife?

The sister's husband inherits half of her estate, and the remainder goes to the brother by 'asabah (agnatic inheritance). Therefore, the estate cannot be disposed of without the husband's consent. If money is handed over to a merchant to invest it in exchange for a specified share of the profit, this is called Mudarabah (definition) or Qirad, and it is permissible if the profit is not fixed at a constant amount. However, if the merchant is obligated to pay a fixed monthly amount, whether the money gains or loses, then this is a فاسد (invalid) Qirad. In the case of an invalid Mudarabah, all the profit belongs to the owner of the money, and the worker is entitled to an 'اجرة المثل' (customary wage) as estimated by experts, or 'ربح المثل' (customary profit) as some jurists have chosen. There is no objection to giving the profits from the Mudarabah as charity on behalf of the sister.