The 27-year-old and team-mates Guido Petti and Santiago Socino were suspended for racist comments posted on social media between 2011 and 2013.
Matera was reinstated two days later at the beginning of December.
But the Argentine Rugby Union now says his captaincy is contingent on the completion of an educational programme.
The UAR had previously said suspension was "unnecessary", describing the social media posts as "imprudent and immature".