Clause 1
Each house shall be the judge of the elections, returns, and qualifications of its own members, and a majority of each shall constitute a quorum to do business; but a small number may adjourn from day to day, and may be authorized to compel the attendance of absent members, in such manner, and under such penalties as each house may provide.
Clause 2
Each house may determine the rules of its proceedings, punish its members for disorderly behaviour, and, with the concurrence of two thirds, expel a member.
Yep, they can certainly operate with a simple majority.
And, as stated in clause 2, they can also choose to operate with other than a simple majority.
The Founding Fathers were not enamored of "simple majorities", which is why so many actions laid out in the Constitution require "supermajorities".