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Mathews remains at the Headingley crease on an impressive 149 not out, with Sri Lanka set to begin the final session of the day on 426-8 - 318 runs ahead of the hosts.
Mathews and Herath's eighth-wicket partnership was worth 149 when confusion led to Herath being run out by Joe Root.
Shaminda Eranga will join Mathews at the crease when the teams resume for the final session of the day.
Liam Plunkett is the pick of England's bowlers with four wickets for 101 runs from his 28 overs.