In one of my textbooks, it was asked how many possible reduced row echelon forms exist for a 3x3 matrix. I found this (Exhaustively) to be 8.

I then thought to myself, how many reduced row echelon forms are there for a n x n matrix? How many for a m x n matrix? I tried to find a logical way to answer this, but my work gets messy very quickly. For n x n I find that I have to begin by expressing it as a sum of sums of sums of...

Is there an easier way to find the number of different forms (Than Exhaustively)?

Many thanks,