Matthew was the fourth round, 84th overall selection of the Buffalo Sabres in the 1992 NHL Entry Draft.
In his first full season with the Buffalo Sabres the NHL the Ottawa, Ontario native led the league with 335 minutes in penalties.
After a run of almost five seasons in the Buffalo organization, Barnaby was dealt to the Pittsburgh Penguins before he was shipped off to Tampa Bay. Always noticeable when on the ice, Barnaby became a fan favorite in Tampa Bay with his feisty play and give all attitude. In 2001-02, he was off to the New York Rangers. Barnaby jumped from Colorado to Chicago to Dallas over three seasons. As a member of the Stars, he would suit up for 39 games with the club before suffering a concussion in a January 9, 2007 in a game against the Coyotes. The concussion Barnaby suffered caused him to miss the remainder of the season and eventually end his career after 834 career NHL games played.