How many distinct matrices are there of a given size, up to row equivalence?

If we define the equivalence relation ~ for matrices $A$ and $B$ such that $A$~$B$ iff $A$ is row equivalent to $B$, then how many equivalence classes does this impose on the set of all matrices? More specifically, what is the cardinality of the set of all equivalence classes? Does the cardinality change if we only look at matrices of a certain size? Has this problem already been solved?

• You don't mention if your matrices are square or not. Nevertheless, there is an infinity of classes because the class of $A$ is characterized by its set of solutions to $AX=0$, i.e., the kernel of $A$, and there are an infinity of possible kernels. See (en.wikipedia.org/wiki/Row_equivalence). – Jean Marie Sep 11 '16 at 22:12
• I didn't think it mattered, since matrices of different sizes cannot be row equivalent. – Zachary F Sep 11 '16 at 22:14
• Of course, but why I meant was different : We fix $m,n$, and consider only $m \times n$ matrices. Do you take $m=n$ ? – Jean Marie Sep 11 '16 at 22:19