Now Matej lives here should also be correct.

I have great difficulties which class the verbs belong to! "bydlit" seems to be class 4. But why, the stem of is byd-? I have the same problems with "jmenovat, spat, psat" and some more verbs. How can I find the right class of a verb in the table given???

Any native English speaker (like me) will agree that "Now Matej lives here" is perfectly normal and should be accepted as this translation.