proportions of invertible matrices

Let GL(d,R) denote the invertiblePlanetmathPlanetmathPlanetmath d×d-matrices over a ring R, and Md(R) the set of all d×d-matrices over R. When R is a finite field of order q, commonly denoted GF(q) or 𝔽q, we prefer to write simply q. In particular, q is a power of a prime.

Proposition 1.

The number of d×d-matrices over a GF(q) is qd2. When a matrix is invertible, its rows form a basis of the vector spaceMathworldPlanetmath GF(q)d and this leads to the following formula


(Refer to order of the general linear groupMathworldPlanetmath. (

Now we prove the ratio holds:


Corollary 2.

As q with d fixed, the proportion of invertible matrices to all matrices converges to 1. That is:

Corollary 3.

As d and q is fixed, the proportion of invertible matrices decreases monotonically and converges towards a positive limit. Furthermore,


By direct expansion we find


So setting a0=1 and


for all i, we have


As ai0, aiai+1 for all i and ai0 as i, we may use Leibniz’s theorem to conclude the alternating seriesMathworldPlanetmath converges. Furthermore, we may estimate the error to the N-th term with error within ±aN+1. Using N=2 we have an estimate of 1-1/q with error ±1q2(q-1). Since q2 this gives 1/2 with error ±1/4. Thus we have at least 1/4 chance of choosing an invertible matrix at random. ∎

Remark 4.

q=2 is the only field size where the proportion of invertible matrices to all matrices is less than 1/2.

Acknowledgements: due to discussions with Wei Zhou, silverfish and mathcam.

Title proportions of invertible matrices
Canonical name ProportionsOfInvertibleMatrices
Date of creation 2013-03-22 15:57:35
Last modified on 2013-03-22 15:57:35
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 16
Author Algeboy (12884)
Entry type Theorem
Classification msc 15A33
Synonym proportions invertible linear transformations