DETERMINANTS AND RANKS OF RANDOM MATRICES OVER Z_{m}

RICHARD P. BRENT AND BRENDAN D. MCKAY

Abstract

Let Z_{m} 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 Z_{m} has a specified m-rank and, if it is square, a specified determinant. These results were previously known only for prime m.

Comments

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

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. rpb101.

(Brent) Centre for Mathematical Analysis, 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, m-rank, singularity, determinant, ring of integers, Gaussian elimination.

Received 24 September 1985. Copyright \begin{matrix}c \\ ◯ \end{matrix} 1987, Elsevier Science Publishers B. V. (North-Holland). Comments \begin{matrix}c \\ ◯ \end{matrix} 1993, R. P. Brent.

rpb094a typeset using AMS-L^{A}TEX