"Match" is acceptable in this context although the far more common verb in the U.S. is "check." Your sentence, however, is not highly idiomatic in other respects.
After solving the problems, match your answers with those in the answer key to ensure that your answers are correct
is pefectly acceptable, but much less frequent than would be
After solving the problems, check your answers against those in the answer key to ensure that your answers are correct
In general, however, the verb "match" would not be used if you confirmed by doing something more logically demanding than using an answer ley. In those cases, the verbs used would usually be "check," "verify," "validate," or "confirm" rather than "match," which simply means verifying that two items are the same.