Absolutely.
If Matthew came first, then when Mark and Luke came across that passage, they either both independently decided to make up a name for the synagogue leader and both chose Jairus (unlikely, I'm sure you will agree), or they had a second source with the name in.
Thanks for clarifying.
I also notice that Mark says Peter, James and John the brother of James went into the house, but the other disciples did not.
This could suggest that one of those three gave Mark the information about what Jesus said to the girl. With Markan priority, that would mean Matthew dropped both the name Jairus, the names of the three who went into the house, and the words said to the girl. He would also have dropped other details such as the girl dying while they were on their way, and the age of the girl, and giving her food after she got up.
(Incidentally, we can assume that Luke quotes from either Matthew or Mark or both, as he states that he has investigated everything and that many people before him have written accounts)
On the other hand, if Mark was the first gospel written, we can explain the situation by assuming Matthew just copied him and Luke copied him but dropped the name. There's no need to postulate a second source. Ockham's razor tells us we should prefer the explanation in which Mark comes first in this instance.
[Matthew doesn't include the name, Luke does]
But as shown above, he didn't just drop the name.
Whatever the case, the mention of flute players by Matthew suggests he is (or has access to) an independent eyewitness, which is interesting from the pov of authenticity.