The Interior Ministry, which is in charge of police, said the shootout was in Fayoum, an oasis province southwest of Cairo and a traditional Brotherhood stronghold.
It said two people who were with the man managed to escape, leaving their SUV behind.
The ministry said the three belonged to the Brotherhood's breakaway faction Hasm, or "decisiveness."
The Brotherhood won a series of elections in Egypt following the 2011 uprising that toppled longtime autocrat Hosni Mubarak, and Mohammed Morsi, a senior Brotherhood leader, became Egypt's first freely elected president the following year.
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The latest crackdown on suspected militants came after an Islamic State affiliate in the Sinai Peninsula -- which is not affiliated with the Brotherhood -- killed 23 Egyptian soldiers in an attack on an outpost Friday.
Since then, the security forces have killed at least 22 suspected militants in shootouts, 14 of them at a training camp in the desert near the Suez Canal city of Ismailia.
Police killed two suspected members of Hasm in a Cairo suburb over the weekend, and yesterday police raided what they said was a hideout for new IS recruits in the Assiut province south of Cairo, killing six of them in a shootout.
Egypt has for years been fighting Islamic militants in Sinai, but the insurgency has expanded and grown deadlier since Morsi's ouster, with some of the attacks now taking place on the mainland.
The Friday attack was the deadliest in two years in Sinai. Funerals for the victims, who included a Lieutenant colonel, attracted thousands of mourners.
Images of the funerals dominated local television and newspaper coverage over the weekend amid a wave of nationalist fervor and calls for retribution.