I know we can multiply a matrix $A$ on the left by some elementary matrix $E$ to get $EA$, which corresponds to an elementary row operation. This preserves a lot of things, such as rank, invertibility, null space, etc.
However, I'm wondering what happens if we try to insert elementary row operations in between a product of two matrices $AB$. For example, something like $AEB$. Does this still preserve things? E.g. does it preserve the rank, invertibility, null space, etc. of $AB$?