Whoose stick?

When it came time to do the stick-snake trick before Paroh, the pasuk indicates that Moshe told Aharon to throw down his stick. So really Aharon’s stick did the job, not that famous fancy Matteh that Moshe brought from Midyan. Why was this? (See Kli Yakar) We suggested that so people wouldn’t think that there was magic in the stick itself, they used different sticks each time. Another suggestion was that Aharon might have been hurt had only Moshe’s stick been used, so they used his stick too.