Holloway, "After you..."

Why does Holloway say "After you", letting his lady walk into the alien pyramid first?

Usually films do this as a movie trope, where the guy is too scared to enter so he plays off the "gentleman" card to have the woman enter into the scary place first..with a wink and a nod. Having the female character hit the male, and then walking into said situation bravely.

But he seemed to be doing it out of politeness here rather than as a joke. Was it some right of passage? Like an honor to be the first one to enter? It's always struck me off. His noncliché way of using the cliché line...