And he claims that keeping the man he calls Mr Celtic at the club can inspire the team to overhaul Rangers in the SPL title race.
Brown finally signed a new four-year contract last Friday, which coincided with the captain's return to fitness following an ankle injury.
And Matthews insists Brown's presence will provide the kind of leadership and drive that can increase the team's momentum as they seek to reclaim the title from Rangers, because the players always raise their game when he is in the side.
"It's given us all a massive boost," the full-back said. "He really is Mr Celtic, he lives and breathes it. When he's playing, everyone just seems to add an extra 10%, so it's a huge plus that he's signed a new contract."
Brown came off the bench at Tannadice yesterday, as Celtic secured a 1-0 win with a comfortable display against Dundee United.
The victory was Celtic's fifth consecutive win in the SPL, a run that has seen them close the gap behind Rangers to four points.
A key contributor to that sustained form has been the increasing steadiness of Celtic's defence, with only two goals conceded in that time.
The clean sheet against United was their ninth in the league this season and Matthews believes that what was once a problem area of the side for manager Neil Lennon is now a far more dependable unit. He said: "We've got to know each other better, we've been playing with each other for longer now and we've starting to understand each other's games.
"We're full of confidence and it was nice to get a clean sheet. Our performance, especially in the first half, was very good against United.
"The defence and the goalkeeper were first-class and with a bit more luck we might have taken more chances. But we got the three points, that's the main thing."
Matthews was switched to left-back from his usual position on the right side of the defence, but he is happy to be seen as a versatile player so long as it means that he stays in the starting line-up.
Although Mikael Lustig, the Swedish right-back, is joining Celtic in January, Matthews welcomed the competition and insisted that he sees his short and long-term future at Parkhead, despite being linked with big-money moves to England.
"I'm happy at Celtic and it's good that we're getting competition for places," the 19-year-old said. "When Lustig comes in, hopefully I can keep playing well and keep my place in the starting line-up.
"It's good for the team that he's coming here. I obviously prefer playing right-back, but I'm just as comfortable on the left.
"If I can get a run in the team, that's fine."
Meanwhile, Neil Lennon has abandoned plans to invite former Manchester United striker Danny Webber for a trial at Lennoxtown.
And reports that Danish teenage central defender, Jacob Dehn Andersen, was on his way to Glasgow have also been dismissed by the Hoops boss.