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My KD_55XG8196 TV does not support AirPlay and cannot play .mkv files, Is there a sony app to enable these capabilities?
If Airplay is not supported how do you expect a Sony app can enable it? You may find alternative on Google play store
MKV container are supported, codec inside may not. You can check yourself which one are supported in the TV user's manuale https://helpguide.sony.net/tv/gaep1/v1/en/04-01-03_03.html?hl=codec