HTML5 video can play .mkv files?
HTML5 doesn't support any video formats, or rather HTML5 doesn't specify what formats browsers should support. It's up to the browsers to decide which formats they choose to support. Apparently Chrome plays .mkv, but I wouldn't be surprised if other browsers didn't play the same file.
How to play mkv files using HTML5 video tag?
WebM uses a variant of the Matroska container so it's not too surprising that some browsers are able to play some .mkv files. It's likely that you'd need to set the content type header to something that the browser recognises (eg. video/webm) if it was going to work at all.
However, that's only part of the story. The Matroska container is just that, a container. Inside the container are video and audio objects which have been encoded with particular codecs, if the browser you're attempting to play the file with doesn't support those codecs then they won't be able to play the file even if they can parse the container format. If you want wide compatibility then your best option is to transcode to one of the more common container/codec combinations such as MP4/h264 or WebM/VP8.
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