Consider the following probability measure on the integers concentrated around $0$: the probability of drawing $0$ is $\frac{1}{2}$, of drawing ($1$ or $-1$) is $\frac{1}{4}$ split evenly among the two choices, of drawing ($2$ or $-2$) is $\frac{1}{8}$ and so forth. Call this measure $\mu$.

Now consider an $n \times n$ matrix $M$ whose entries have been drawn in a $\mu$-iid fashion. Define a $\mu$-random row operation on $M$ as follows: pick two numbers $p,q$ randomly according to the uniform distribution on $\lbrace 1,2,\ldots,n \rbrace$ and also pick an integer $m$ according to $\mu$. Then, we perform the row operation that replaces the $p$-th row of $M$ by itself plus $m$ times the $q$-th row. Similarly define a $\mu$-random column operation.

What does the distribution of entries in $M$ look like as the number of $\mu$-random row and column operations goes to $\infty$?

## Background

I should confess that this question does not arise from my own research, but rather from watching the progress of some undergraduates taking linear algebra who have not quite mastered the Gaussian elimination algorithm yet. Their strategy for row reduction seemed to involve blindly performing row operations without caring for pivots or GCD's or anything of the sort. In order to incur a minimal computational burden, they seemed to prefer small scalar multiples of rows, so I chose $\mu$ to be concentrated on the small integers. What are the odds that they will get lucky with this blind strategy and end up with a matrix largely populated with zeros?