Five soldiers were killed or wounded in the operations last week, according to the online video statement Saturday, which did not specify the number of military fatalities.
The video, replete with a dramatic score, said troops had killed three "extremely dangerous" militants as well as 16 others in precision airstrikes on their "terrorist hideouts".
"Two officers, a non-commissioned officer and two soldiers were killed and wounded in the operations", the statement said, according to AFP.
Troops had recovered troves of automatic rifles, hand grenades and rocket-propelled grenades in possession of the militants, the army added.
Security forces have been battling a long-running extremist insurgency in the Sinai Peninsula in Egypt's northeast that is spearheaded by a local affiliate of the Daesh terrorist group.
The fighting intensified after the military's 2013 ouster of president Mohamed Morsi, who was replaced by President Abdel Fattah al-Sisi.
In February 2018, authorities launched a nationwide operation against militants, focused on north Sinai.
Around 970 suspected militants have been killed in the region along with dozens of security personnel, according to official figures.