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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 $AB$?

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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.

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  • $\begingroup$ 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. $\endgroup$ – twosigma May 30 '20 at 17:12
  • $\begingroup$ @twosigma Yes that is correct $\endgroup$ – Ben Grossmann May 30 '20 at 17:13

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