MS Dhoni's cameo in vain as five-time champions dealt thrid loss but hold on to third place in the standings
England's Rose equalled the 36-hole tournament record with a two round 15-under par total of 129. Japan's Matsuyama was alone in second after a six-under par 66 for 132.
Overnight leader Jon Rahm of Spain headed a trio sharing third place on 134, while 14-time major winner Tiger Woods found himself 11 adrift after a sometimes frustrating two-under par 70.
Rose started the day one shot behind Rahm and got going with three birdies in a row at the fourth, fifth and sixth on the South Course, one of two in use over the first two days of the tournament
After his lone bogey of the day at the seventh Rose saw a birdie try at eight lip out, but he moved to the lead at 12-under with a two-putt birdie at the par-five ninth, then rolled in a two-footer for a birdie at the 10th.
Rose birdied the par-five 13th and rolled in a 24-footer for birdie at 16 to pad his lead.
"The job was done today," said Rose, who has hit 16 of 18 greens in regulation in each of the first two rounds. "It's really nice to go out there and place a positive round and feel good on it. I can build on it hopefully on the weekend."
American Billy Horschel, who raced into contention with five birdies in the space of six holes on the front nine, saw his challenge fade with three bogeys on the back nine on the way to a four-under par 68.
That left him tied with Ryan Palmer (67) and Rahm, who had two birdies and two bogeys in his even par round on the South Course - a far cry from his sparkling 62 on Thursday.
Matsuyama's six-under effort on the South Course included an eagle from off the green at the par-five sixth.
He teed off on 10 and made the last of his five birdies at the par-three eighth.
Northern Ireland's world number eight Rory McIlroy, playing the event for the first time, climbed the leaderboard with a seven-under par 65 on the North Course, where he rolled in an eagle putt at his first hole, the 10th, and holed out for an eagle at the at 16.
McIlroy added four birdies with just one bogey for an eight-under par total of 136 that left him tied for ninth.
"All in all a good round of golf," McIlroy said. "At least I have a good shot going into the weekend."
Woods couldn't say the same. Needing to go low after an opening 70 on the South Course, Woods settled for the same on the North.
"I just couldn't get a putt to fall," said Woods, although he added that the problem wasn't poor putting.
"A lot of lip-outs today and a round that could easily have been five- or six-under par just didn't materialize."
Woods was even for the day through his first nine holes after a double-bogey at the 18th - where he was plugged in a bunker - offset two earlier birdies.
That left him flirting with the cut line before back-to-back birdies at the third and fourth, the second courtesy of a chip-in. But Woods wouldn't find another birdie the rest of the way, unable to take advantage of a superb drive at the fifth and watching his chip at the sixth hit the stick and bounce away.
"It's going to have to be a low and special weekend to have a chance," said Woods, whose eight victories at Torrey Pines include seven Farmers Insurance Open titles and a memorable 2008 US Open triumph - the most recent of his 14 major titles.
MS Dhoni's cameo in vain as five-time champions dealt thrid loss but hold on to third place in the standings
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