Matthias was born in Frankfurt, Germany, and educated at Rome University and at the Federal Institute of Technology, Zurich, where he obtained his PhD in 1943. He moved to America in 1947, became naturalized in 1951, and, after a brief period at the University of Chicago, joined the staff of the Bell Telephone Laboratories. In 1961 he returned to academic life when he was appointed professor of physics at the University of California, San Diego.

Matthias carried out extensive work on superconducting materials. In the early 1950s no existing theory of superconductivity allowed deductions as to which metals were superconductors and at what temperature – their transition point – they became so. Consequently Matthias set out to find such materials by experiment, testing thousands of alloys in the hope that some kind of pattern would emerge. He found that superconductivity depended on the number of outer electrons in the atom; substances with five or seven valence electrons most readily became superconductors and that they had transition points furthest above absolute zero. The crystal structure of the solid was another important factor. As a result of these empirical observations, Matthias and his collaborators were able to make new superconducting materials, including a niobium–germanium alloy with a transition temperature of 23 K. Matthias also did important work on ferroelectric materials.