Can a woman say “Ameen” out loud when she is praying with her husband at home?

Firstly: 

Saying “Ameen” is Sunnah for every worshipper when he finishes reciting al-Faatihah. 

Al-Nawawi (may Allaah have mercy on him) said in al-Majmoo’, 3/371: 

Saying “Ameen” is Sunnah for every worshipper after al-Faatihah, whether he is leading the prayer or is following an imam, or praying alone, for men, women and children, whether they are praying standing up, sitting down or lying down (i.e., if they have an excuse), whether they are praying an obligatory prayer or a naafil prayer, whether the prayer is one in which Qur’aan is recited silently or out loud. There is no difference of opinion concerning this among any of our companions. 

Secondly: 

Women are not permitted to raise their voices when they are present with men who are not their mahrams. Hence the Prophet (peace and blessings of Allaah be upon him) forbade women to say “Subhaan-Allaah” during prayer if they want to alert the imam to something; rather they should alert him by clapping. 

It was narrated from Sahl ibn Sa’d al-Saa’idi that the Messenger of Allaah (peace and blessings of Allaah be upon him) went to Bani ‘Amr ibn ‘Awf to reconcile between them. (In the meantime) the time for prayer came, and the muezzin came to Abu Bakr and said, “Will you lead the people in prayer, and should I say the iqaamah?” He said, “Yes.” So Abu Bakr started to lead the prayer. Then the Messenger of Allaah (peace and blessings of Allaah be upon him) came when the people were still praying, and he passed through the rows until he stood in the first row. The people clapped but Abu Bakr used not to glance sideways whilst he was praying. When the people’s clapping continued, he turned and saw the Messenger of Allaah (peace and blessings of Allaah be upon him). The Messenger of Allaah (peace and blessings of Allaah be upon him) gestured to him to stay where he was, so Abu Bakr (may Allaah be pleased with him) raised his hands and praised Allaah for what the Messenger of Allaah (peace and blessings of Allaah be upon him) had told him to do. Then Abu Bakr stepped back until he was level with the row and the Messenger of Allaah (peace and blessings of Allaah be upon him) came forward and led the rest of the prayer. When he finished he said, “O Abu Bakr, what kept you from staying put when I told you to?” Abu Bakr said, “It was not right for the son of Abu Quhaafah [referring to himself] to lead the Messenger of Allaah (peace and blessings of Allaah be upon him) in prayer.” The Messenger of Allaah (peace and blessings of Allaah be upon him) said, “ Why did you clap so much? If something happens to anyone during his prayer, let him say Subhaan Allaah, for if he says Subhaan Allaah, people will take notice. Clapping is only for women.” 

(Narrated by al-Bukhaari, 652; Muslim, 421) 

Ibn Hajar said: 

He did not let women say Subhaan Allaah because they are commanded to lower their voices in prayer in all circumstances, because of the fear that their voices may cause fitnah (temptation). He forbade men to clap because this is something for women. 

Fath al-Baari, 3/77 

This prohibition applies if there are non-mahram men present. But if she is with a group of women or with men who are her mahrams, there is nothing wrong with her reciting and saying “Ameen” out loud. 

Ibn Qudaamah (may Allaah have mercy on him) said in al-Mughni (3/38): 

A woman may recite out loud in prayers where Qur’aan is recited out loud, but if there are men present, she should not recite out loud, unless they are her mahrams, in which case she may do that. 

Al-Nawawi said in al-Majmoo’ (3/390): 

With regard to women, most of our companions said that if she is praying on her own, or in the presence of other women or men who are her mahrams, she may recite out loud, whether she is leading other women in prayer or praying alone. But if she is praying in the presence of non-mahram men she should recite silently… This is our opinion… al-Qaadi Abu al-Tayyib said: The ruling on saying Allaahu akbar in prayers where Qur’aan is recited out loud and prayers where it is recited silently is the same as the ruling on reciting Qur’aan. 

The ruling on saying Ameen in prayers where Qur’aan is recited out loud and prayers where it is recited silently is the same as the ruling on reciting Qur’aan. 

Ibn Qudaamah (may Allaah have mercy on him) said in al-Mughni (2/162): 

It is Sunnah for the imam and the members of the congregation to say it (Ameen) out loud when Qur’aan is recited out loud, and to say it silently in prayers where Qur’aan is recited silently. 

Al-Nawawi said in al-Majmoo’, 3/371: 

If the prayer is one in which Qur’aan is recited silently, then the imam and others should say Ameen silently, following the manner of recitation. If it is a prayer in which Qur’aan is recited out loud, it is mustahabb for the members of the congregation to say Ameen out loud. There is no difference of opinion on this point. 

In conclusion: 

It is permissible for a woman to recite Qur’aan and say Ameen out loud in prayer, unless she is praying in the presence of a non-mahram man, in which case she should say it silently. 

And Allaah knows best.