If your Muslim brother argues with you, then strikes you, do you strike him back?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

Allah, The Wise Lawmaker, has authorized the wronged person to return the offense in the same way and to take his right.
Allah Says (what means):

• {So whoever has assaulted you, then assault him in the same way that he has assaulted you. And fear Allah and know that Allah is with those who fear Him.} [Quran 2:194]

• {And the retribution for an evil act is an evil one like it.} [Quran 42:40]

However, pardoning the wrongdoer/attacker is better, and greater in reward. Allah Says (what means): {… but whoever pardons and makes reconciliation - his reward is [due] from Allah.} [Quran 42:40]

Know that showing patience towards the other's harm, and suppressing anger are two characteristics of the pious, who will inherit Paradise by Allah's Will, Grace, and Pardon on the Day of Judgment.

Allah Says (what means): {And hasten to forgiveness from your Lord and a garden as wide as the heavens and earth, prepared for the righteous. Who spend [in the cause of Allah] during ease and hardship and who restrain anger and who pardon the people - and Allah loves the doers of good.} [Quran 3:133-134]

In the same vein, the Prophet said: “A person who suppresses his anger although he can revenge, Allah will fill his/her heart with contentment on the Day of Resurrection.” [At-Tabarani]

Allah knows best.