Arthur Matthews studied at Sidney Sussex College, Cambridge, where he obtained the degrees Bachelor of Arts (BA) in 1870 and Master of Arts (MA) in 1873. However, at some time he also obtained a Bachelor of Science (BSc) degree from the University of London. He came to the Cape Colony in 1875 as the first lecturer in mathematics and physical science at St Andrew's College, Grahamstown, and later became vice-principal of its College Department. This department prepared students for some of the examinations of the University of the Cape of Good Hope. During his years at St Andrews he published Notes on chemico-physics and the chemistry of the non-metals (Grahamstown, 1886).
Matthews remained at St Andrew's until 1904. On 31 May that year Rhodes University College was established and its first four professors appointed from the staff of St Andrew's College. One of these was Matthews, who became professor of mathematics. While still at St Andrew's he had started training students in the theory of surveying and he continued these classes at Rhodes. His course in surveying was so popular that 40 of the first 50 students who enrolled at Rhodes took it, even though it included many subjects and required much work. He retired from his post in November 1910 and died the next year.
Matthews was a foundation member of the South African Geological Association, which was established in Grahamstown in 1888 but soon ceased to function. He was also a foundation member of the South African Association for the Advancement of Science in 1902, but did not remain a member for long. In 1902 he compiled a Register of S. Andrew's College, Grahamstown, from 1855 to 1902, which was published in Cape Town.