ON DETERMINANTS OF RANDOM SYMMETRIC MATRICES OVER Z_{m}

RICHARD P. BRENT AND BRENDAN D. MCKAY

Abstract

We determine the probability that a random n × n symmetric matrix over {1, 2, …, m} has determinant divisible by m.

Comments

Only the Abstract is given here. The full paper appeared as [2]. For related work on random unsymmetric matrices, see [1].

References

[1] R. P. Brent and B. D. McKay, “Determinants and ranks of random matrices over Z_{m}”, Discrete Mathematics 66 (1987), 35–49. MR 88h:15042. Also appeared as “Determinants and ranks (mod m) of random integer matrices”, 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 Z_{m}, Ars Combinatoria 26A (1988), 57–64. MR 90g:05015. Also appeared as Report TR-CS-88-03, CSL, ANU, February 1988, 8 pp. (Retyped in L^{A}TEX 1998.) rpb101.

(Brent) Computer Sciences Laboratory, Australian National University

E-mail address: rpb@cslab.anu.edu.au

(McKay) Computer Science Department, Australian National University

E-mail address: bdm@cs.anu.edu.au

1991 Mathematics Subject Classification. Primary 05A15, 15A52; Secondary 12C99, 15A03, 16A42, 16A44. Key words and phrases. Random matrix, random symmetric matrix, singularity, determinant, ring of integers,

Gaussian elimination.

Copyright \begin{matrix}c \\ ◯ \end{matrix} 1988, Charles Babbage Research Centre. All rights reserved. Any individual may photocopy one copy of any paper in this volume for purposes of study or research. There is no charge.

Comments \begin{matrix}c \\ ◯ \end{matrix} 1993, R. P. Brent.

rpb101a typeset using AMS-L^{A}TEX.