An algorithm is given that computes the determinant of an n x n matrix with entries from an arbitrary commu-
tative ring in O(n^3*sqrt(n)) ring additions, subtractions, and multiplications; the "soft-O" О indicates some missing logn factors. The exponent in the running time can be reduced further by use of asymptotically fast matrix multiplication. The same results hold for computing all n^2 entries of the adjoint matrix of an n x n matrix with entries from a commutative ring.