Determinants and ranks of random matrices over Zm

Let Zm be the ring of integers modulo m. The m-rank of an integer matrix is the largest order of a square submatrix whose determinant is not divisible by m. We determine the probability that a random rectangular matrix over Zm has a specified m-rank and, if it is square, a specified determinant. These results were previously known only for prime m. CommentsOnly the Abstract is given here. The full paper appeared as [1]. For related work on randomsymmetric matrices, see [2]. References[1] R. P. Brent and B. D. McKay, “Determinants and ranks of random matrices over Zm”, Discrete Mathematics66 (1987), 35–49. MR 88h:15042. Also appeared as “Determinants and ranks (mod m) of random integermatrices”, Report CMA-R25-85, CMA, ANU, August 1985, 17 pp. rpb094.[2] R. P. Brent and B. D. McKay, On determinants of random symmetric matrices over Zm, Ars Combinatoria26A (1988), 57–64. MR 90g:05015. Also appeared as Report TR-CS-88-03, CSL, ANU, February 1988, 8 pp.rpb101. (Brent) Centre for Mathematical Analysis, Australian National UniversityE-mail address: [email protected](McKay) Computer Science Department, Australian National UniversityE-mail address: [email protected] 1991 Mathematics Subject Classification. Primary 05A15, 15A52; Secondary 12C99, 15A03, 16A42, 16A44.