Please could you tell me which name option is correct?
A) The concert was held in the Cultural Centre named after P.I.Tchaikovsky.
B) The concert was held in the P.I.Tchaikovsky Cultural Centre.
Is it correct to use the definite article before the name of the centre in both sentences?
Yes, you should use the definite article in both sentences, but in A it should be “cultural centre” because without the “P.I. Tchaikovsky”, it is not a proper name. This is similar to “The concert was held in the city of New York.” “The concert was held in New York City.” - we do not capitalize the first “city”. Although notice it is capitalized in this: "The concert was held in the P.I.T. Cultural Centre, named after P.I.Tchaikovsky.
Thank you very much, Luschen!