The compounds of a matrix play an important role in several branches of mathematics, such as in algebraic geometry. In the present paper, Mahler discusses applications of such matrices to the theory of convex bodies and to the geometry of numbers. \par Reprint of the author's paper [Proc. Lond. Math. Soc. (3) 5, 358--379 (1955; Zbl 0065.28002)].