Both of these are real words, but they have quite different meanings. The difference isn’t complicated but it can be hard to remember which one is correct, especially if you’re in an exam situation.
to complement (v)
This means to go with, or to match. ‘Her handbag complements her shoes’.
As a noun, complement refers to the number of people on a vessel. ’The HMS Pinafore has a complement of 80’. (Means there are 80 people on board the ship.)
1. To say positive things about someone. ‘He makes great websites!’ ‘Her accent is so sexy.’ You could even compliment someone by saying, ‘Her handbag complements her shoes’. In other words, the woman has great taste and dresses well. That’s a compliment.
2. To do something free of charge. ‘Please take one of these complimentary brochures.’ How much do these drinks cost? / Nothing, they’re complimentary.
“Your eyes compliment your face.” WRONG. This means that your eyes can talk.
“Your eyes complement your face.” Correct.