The correct answers are I boarded the wrong flight to Antarctica by mistake. and To make matters worse, I had packed only beach clothes.

Sometimes the rule is that there is no article at all! These are called "zero article" fixed expressions.

• "By mistake" never takes an article (never "by a mistake" or "by the mistake").

• "To make matters worse" is a set phrase that does not use "the" before "matters." Adding an article breaks the idiom!