Important advice to the spouses concerning discussions between them
Question

Sometimes my wife wants to discuss an issue with me and she has a valid point but I do not want to discuss it with her. That leads to her getting upset and it ends in an argument. Despite that I do not admit that it was my fault until she comes and tries to reconcile me. Please note that she knows that I am the one who is in the wrong.
Praise be to Allah.What you have to do is treat your wife kindly. Part of kind treatment is listening properly to what she is saying and responding properly. Being right is not limited to men; it may be your wife’s view that is correct and her suggestions and advice may be good. What is preventing you from letting her express her views and discussing them with her in a friendly manner? 
Remember how the Prophet (blessings and peace of Allah be upon him) listened to the advice of his wife Umm Salamah (may Allah be pleased with her) and adopted it in a matter of great importance. That was during the well-known Treaty of al-Hudaybiyah, when she suggested to him that he should go out and not speak to anyone among his Companions until he had slaughtered his hadiy (sacrificial animal) and shaved his head. Our Prophet Muhammad (blessings and peace of Allah be upon him) did as his wife (may Allah be pleased with her) suggested and there was a great deal of good in that. 
Hence we advise you to be the one to start a discussion between you and your wife and open your heart and mind to what she has to say, and listen attentively. Then you can either accept what she says and thank her, or reject it gently whilst also thanking her. Do not forget that she is your life partner and the one who is helping you to raise your children and organise your household. So it is not appropriate to close the doors of discussion to her. You should not be too proud of your opinion and or feel that you do not need to discuss with her and hear her opinion. Do not neglect the words of your Prophet Muhammad (blessings and peace of Allah be upon him): “The best of you is the one who is best to his wife.” Narrated by at-Tirmidhi, 3895; classed as saheeh by Shaykh al-Albaani in Saheeh al-Jaami‘, 3314. Over and above that, do not forget the words of Allah (interpretation of the meaning): “And live with them honourably” [an-Nisa’ 4:19]. Discussion between the spouses increases the love between them and strengthens the bond between them; it also helps to reach the right decision in matters of married life and household and family affairs. 
Remember that the consequences of this attitude towards your wife may backfire on you; she may refuse to give you any advice so long as you do not accept what she offers you and is forced to agree with you even if you are wrong. Thus you will be depriving yourself of a great deal of good and beneficial advice. It suffices us to say to you: This attitude, in which you insist on your opinion, do not admit your mistakes and do not acknowledge that your wife could be right is the very essence of arrogance, as the Prophet (blessings and peace of Allah be upon him) defined it. 
It was narrated from ‘Abdullah ibn Mas‘ood (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) said: “No one will enter Paradise who has an atom’s weight of arrogance in his heart.” 
A man said: What if a man likes his garment to look nice and his shoes to look nice? 
He said: “Allah is Beautiful and loves beauty. Arrogance is rejecting truth and looking down on people.” Narrated by Muslim, no. 91 
What is meant by rejecting truth is denying it out of arrogance and pride. 
Looking down on people means scorning them and thinking little of them. 
Don’t you see that this is the essence of what you are doing with your wife? 
Are you content to have this attitude and hence this fate may be yours? 
May Allah protect you from that, O slave of Allah. 
It is not the attribute of the wise and mature man to insist when he is in the wrong that his wife should be the one to apologise. Rather he is the one who should hasten to admit his mistake, apologise for it and ask his wife to forgive him. And her duty is to apologize to her husband if she is the one who is in the wrong. It is in this way that married life takes a correct course and lasting love and compassion are strengthened between the spouses. Abu’d-Darda’ (may Allah be pleased with him) said to his wife: “If you see me angry, try to calm me down, and if I see you angry I shall try to calm you down, otherwise we cannot live together.”  
You could also benefit by reading the book The Muslim Home: 40 Recommendations, which you can find in the Articles and Books section of this website. 
We ask Allah to guide you and your wife to that which He loves and is pleased with, and to bring you together in the best of ways. 
And Allah knows best.