We find the maximum number of orthogonal skewsymmetric anticommuting integer matrices of order n for each natural number n and relate this to finding free direct summands of certain generic projective modules. While studying composition of quadratic forms, Hurwitz [4] and Radon [6] considered families of orthogonal matrices {A1,3L , A,} satisfying the conditions (1) Ai= -At, i= 1, ,s (2) AiAj = -AjAi, i $ j. DEFINITION. (1) A family of orthogonal matrices satisfying (1) and (2) above will be called a Hurwitz-Radon (H-R)family. If n is a positive integer and n=2ab, b odd, then we write a=4c+d where O0d<4. If we denote by p(n) the number 8c+24 the main theorem of Radon states: THEOREM A [6]. (1) Any H-R family of real matrices of order n has fewer than p(n) members. (2) There is an H-R family of real matrices of order n having exactly p(n)-1 members. In the first section we prove an analogous theorem for integer matrices and in ?II we consider some applications to the study of projective modules. We are indebted to R. Gabel for having furnished us with a copy of his Brandeis thesis. The ideas studied here were inspired by that work and represent a simplification and extension of one part of that thesis. Gabel Presented to the Society, November 25, 1972; received by the editors September 7, 1972 and, in revised form, January 30, 1973. AMS (MOS) subject classifications (1970). Primary 15A36, 15A63, IOJ05, 13C10.