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The Wimbledon champion was pushed to four sets in his opening match on the clay court, but it was a simpler affair this time around as he broke his opponent at the first time of asking.
After taking a 2-0 cushion, games continued by the way of serve, but Matosevic came close to sealing a break of his own until Murray saved three break points to stay on top.
With a one-set lead, the Scotsman was pushed to the limit in the opening game of the second set, but he eventually held his serve.
Murray responded by instantly breaking Matosevic, who continued to push, but it wasn't to be as a further break point for the British number one ensured that he was just one set away from winning the encounter.
It was a battling opening game in the third set as the scoreline was brought back to deuce nine times before Murray broke and then held to take a comfortable lead.
Despite Matosevic's attempt to launch a comeback, it was relatively comfortable for Murray in the end as he grabbed another break to push on and progress into the next round.