"No, I am Matěj."
Translation:Ne, Matěj jsem já.
It is something between and frustratingly complex. Some words, which usually are not verbs, occupy the 2nd position. Other components (not necessarily single words but rather independent pieces) can be arranged in whichever order works for what the known information is, what is being newly communicated, and the things in between.
"Ne, jsem já Matěj." is not a valid order. Even if we removed the "Ne" piece, it would still be wrong. It has to do with the weird order "jsem já" being followed by the verb complement. In general, all orders VERB-SUBJECT PRONOUN-COMPLEMENT are highly suspect in Czech statements, although they work in questions, like "Jí on maso?"
Using "ja" isn't strictly necessary because the verb is conjugated; it is usually a for emphasis or contrast. "Jsem Matej" is a neutral statement identifying himself; "Ja jsem Matej" is an emphatic statement (I am Matej, not someone else - see above on word order) or a contrast when there is more than one subject - "On je Frantisek, a ja jsem Matej."