Computing the Determinant and Smith Form of an Integer Matrix

A probabilistic algorithm is presented to find the determinant of a nonsingular, integer matrix. For a matrix A n n the algorithm requires O n3 5 logn 4 5 bit operations (assuming for now that entries in A have constant size) using standard matrix and integer arithmetic. Using asymptotically fast matrix arithmetic, a variant is described which requires O n2 2 log2 n loglogn bit operations, where two n n matrices can be multiplied with O n operations. The determinant is found by computing the Smith form of the integer matrix, an extremely useful canonical form in itself. Our algorithm is probabilistic of the Monte Carlo type. That is, it assumes a source of random bits and on any invocation of the algorithm there is a small probability of error.