From compliment (noun) +‎ -ary.^[1] Piecewise doublet of complementary.

• (Received Pronunciation) IPA^(key): /ˌkɒmplɪˈmɛnt(ə)ɹi/
• (General American) enPR: kŏm'plĭ-mĕnʹtə-rē, -trē, IPA^(key): /ˌkɑmplɪˈmɛnt(ə)ɹi/

• Audio (US):

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• Homophone: complementary
• Rhymes: -ɛntəɹi, -ɛntɹi
• Hyphenation: com‧pli‧men‧ta‧ry

complimentary (comparative more complimentary, superlative most complimentary)

1. In the nature of a compliment.

   Antonym: uncomplimentary

   a complimentary remark

   • 1877 August 25, Mark Twain, The Letters of Mark Twain, volume 3,2004, page 56:

     But yesterday evening late, when Lewis arrived from down town he found his supper spread, and some presents of books there, with very complimentary writings on the fly-leaves, and certain very complimentary letters, and more or less greenbacks of dignified denomination pinned to those letters and fly-leaves...

2. Free; provided at no charge.

   Synonyms: free of charge; see also Thesaurus:gratis

   complimentary drinks

   • 2005, Vivien Lougheed, Belize Pocket Adventures, page 65:

     In the evenings, Ward and PJ offer guests a complimentary drink. This gesture seems to reinforce the hosts' desire to make everyone feel welcome.

3. With respect to the closing of a letter, formal and professional.
   • 2003, Richard H. Beatty, The Perfect Cover Letter:

     The complimentary closing is the word grouping used to bring the message or text to a close.

• Complimentary and its homophone complementary are frequently confused and misused in place of one another.

• complimentary on Wikipedia.Wikipedia