Is the wife religiously obligated to repay her deceased husband's debt, which is an amount he deposited with her for his mother's account, knowing that the husband's brothers are aware of this and have debts to the deceased?

The wife is not obligated to repay her deceased husband's debt. Rather, the debt is to be paid from his wealth before the distribution of the inheritance, in accordance with the Almighty's saying: [after any bequest he may have made or any debt]. The debt his brothers claim from him is a right he has over them, and they must pay it to his heirs. Regarding the money the mother gave to the husband: If it was for investment and the husband did not squander it or dispose of it without permission, then he is not liable for it if his brothers usurped it. If it was for savings and the husband voluntarily invested it with the mother's knowledge and approval, then he is also not liable for it. If it was a deposit for safekeeping and the husband placed it in the company without her permission, then he is liable for the money due to his unauthorized disposal, and the brothers must return the money to the mother if their knowledge of this is proven.

It is necessary to refer to the Sharia courts to seek evidence, and the mother should advise her sons to abstain from the orphans' money and not to wrong the wife.