Matthews was the son of William and Julie Murfee Matthews. He attended Mountain Brook Elementary School and Birmingham University School and graduated from Marion Military Institute and the Georgia Institute of Technology with a bachelor's degree in electrical engineering. He began working at Southern Natural Gas as an engineer in 1950. He served in the U.S. Navy from 1954 to 1956 and rose to the rank of Lieutenant. After returning to the company he was promoted to president in 1984 and served as executive vice-president before retiring in 1991.
Matthews was elected Mayor of Mountain Brook in 1992 and served one four-year term. He pushed for expansion of sidewalks connecting the city's commercial villages, accepting the city's first-ever federal grant funding. He also pushed for a property tax increase which was defeated by referendum, but managed to maintain a budget surplus each year. He resigned before the end of his term in 1996, allowing Mountain Brook City Council president Margaret Porter to take the office of Mayor.
Matthews was also a long-time elder at 1st Presbyterian Church of Birmingham and helped found the Rushton Child Development Center and the First Light women's shelter in the former Granada Hotel. He served on the boards of Alabama Independent Colleges, YMCA of Birmingham, Highlands Day School, Blue Cross Blue Shield of Alabama, Kiwanis Club of Birmingham, the Metropolitan Development Board and the board of advisors of Birmingham-Southern College.