Matthews attended Elkins High School in Missouri City, where he was a standout offensive lineman for the Elkins Knights high school football team. He was teammates with D. J. Hayden. In his senior season, Matthews was a first-team all-district and all-greater Houston honoree as his team finished the season with a 9–3 record. Matthews was invited to play in the 2010 U.S. Army All-American Bowl and also earned 2009 High School All-American honors by USA Today, Parade, and SuperPrep.
Regarded as a four-star recruit by Rivals.com, he was listed as the No. 7 offensive tackle prospect in the class of 2010. He chose Texas A&M over offers from Oklahoma, Alabama, Arkansas, Oregon, Texas, Texas Tech, Southern California, and Stanford.
As a junior in 2012, Matthews was a first-team All-Southeastern Conference (SEC) selection in recognition of his successful season. He also received first team All-American honors by the Football Writers Association of America. During his senior season, Matthews played left tackle after playing right tackle for the previous three seasons.
As a senior in 2013, Matthews was again a first-team All-Southeastern Conference (SEC) selection.
Matthews was drafted by the Atlanta Falcons with the sixth overall pick in the 2014 NFL Draft. With Luke Joeckel having been selected second overall in the 2013 NFL Draft by the Jacksonville Jaguars, Texas A&M became the first school with consecutive top-10 selected offensive lineman since USC's Tyron Smith and Matt Kalil in 2011 and 2012, respectively.
Matthews was selected in the first round with the sixth overall pick by the Atlanta Falcons in the 2014 NFL Draft. In his first NFL game, Matthews suffered a high ankle sprain in a season-opening win against the Saints. Despite the injury, Matthews started in all but one game in his rookie season. Matthews was rated the worst tackle of the 2014 season by Pro Football Focus. In 2015, Matthews' blocking improved, allowing only 38 quarterback pressures compared to 51 in 2014. That season, Matthews was named as the third most improved player of 2015 by Pro Football Focus.
In the 2016 season, Matthews and the Falcons reached Super Bowl LI, where they faced the New England Patriots. In the Super Bowl, the Falcons fell in a 34–28 overtime defeat.
On April 25, 2017, the Falcons picked up the fifth-year option on Matthews' contract.
Matthews was born in Missouri City, Texas. He is the son of Pro Football Hall of Famer Bruce Matthews and the younger brother of Kevin Matthews, a center who also played for the Aggies. His younger brother, Mike, last played center for the Cleveland Browns. He is also the cousin of Clay Matthews III, a linebacker for the Green Bay Packers, and Casey Matthews, a linebacker who last played for the Minnesota Vikings. He is the nephew of NFL veteran Clay Matthews, Jr., the grandson of NFL veteran Clay Matthews, Sr., and the great grandson of H. L. Matthews.
The Matthews football dynasty family tree: