His book Theory of Matrices (1953) is a standard reference of linear algebra. It has been translated into various languages including a two-volume version in English prepared by Joel Lee Brenner, Donald W. Bushaw, and S. Evanusa.[1][2]George Herbert Weiss noted that "this book cannot be recommended too highly as it contains material otherwise unavailable in book form".[3]
Gantmacher collaborated with Mark Krein on Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems.[4]
In 1939 he contributed to the classification problem of the real Lie algebras.[5]