Can we use row operations in between matrices?

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 $$AB$$?

It preserves invertibility when $$A$$ and $$B$$ are square, but nothing else. Two examples where the properties you mention are not preserved is $$A = B = \pmatrix{1&0\\0&0}, \quad E = \pmatrix{0&1\\1&0};\\ A =\pmatrix{1&0\\0&0}, \quad B = \pmatrix{0&0\\0&1}, \quad E = \pmatrix{0&1\\1&0}.$$ It is true that $$AB$$ and $$AEB$$ will both have column-spaces contained by the column space of $$A$$ and null spaces that contain the null space of $$B$$. Besides that, I don't think much can be said.
• Ok. And the fact that invertibility is preserved can, for example, be seen by the fact that $\det (AB) = \det(A) \det(B)$ is nonzero if and only if $\det(AEB) = \det(A) \det(E) \det(B)$ is nonzero, since $\det(E)$ is always nonzero. – twosigma May 30 '20 at 17:12