If $A$ can be row reduced Echelon to $X$ and $B$ can be row reduced Echelon to $Y$ Is it true that that $AB$ can be row reduced to $XY$?

Question

If $A$ can be row reduced Echelon to $X$ and $B$ can be row reduced Echelon to $Y$ How can I prove that $AB$ can be row reduced to $XY$?

This might sound intuitive but I couldn't prove it

I tried to use induction and assumed that $A,B$ are elementary matrices so they can by row reduced by single row operation but I found it difficult to prove this and $A,B$ need not be square matrix

Your observations are impeccable but I still don't see your point. Try this. Take an arbitrary $2\times 3$ matrix. Can you reduce it with a $2\times 2$? — John Douma, Jan 08 '24 at 13:49
@JohnDouma yes I can make this matrix a roe reduced Echelon by multiplying that $2\times$ $3$ matrix with some of $2\timese$ elementary matrices — Mathematics enjoyer, Jan 08 '24 at 13:53
There you go. Notice that the row operations are independent of the size of the row. So if $A$ and $X$ are $m\times n$ and $B$ and $Y$ are $n\times p$ then you can reduce the first with an $m\times m$ and the second with an $n\times n$ and when you multiply $(m\times m)(m\times n)(n\times n)(n\times p)$, what do you get? — John Douma, Jan 08 '24 at 14:00
@JohnDouma I had a hard time writing a good rigorous proof of that — Mathematics enjoyer, Jan 08 '24 at 14:47
Hard work is good for us. You should have what you need. Good luck. — John Douma, Jan 08 '24 at 14:58
It should be noted that the statement is wrong. If you try to prove it using elementary matrices, you will see that it won't work, essentially because matrix multiplication is not commutative. — leoli1, Jan 10 '24 at 06:03
Row-reducing $A$ to $X$ is finding a matrix $C$ such that $CA=X$ (or maybe it's $AC=X$, I can never remember which, but it doesn't matter as long as I'm consistent). So you have $CA=X$ and $DB=Y$, and you want $EAB=XY$. If $AD=DA$, you win: $XY=CADB=CDAB=EAB$ with $E=CD$. But if $AD\ne DA$, then it seems to me you're out of luck. — Gerry Myerson, Jan 10 '24 at 18:11

score 4 · Accepted Answer · answered Jan 11 '24 at 13:36

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As I understood your question, there is a simple counterexample. Namely, put $A=B=\begin{pmatrix} 1 & 0\\ 0 & 0\end{pmatrix}$ and $X=Y=\begin{pmatrix} 0 & 0\\ 1 & 0\end{pmatrix}$. Then $A$ can be row reduced to $X$ and $B$ can be row reduced to $Y$, but $AB=A$ cannot be row reduced to $XY=0$, because these matrices have different ranks.

answered Jan 11 '24 at 13:36

Alex Ravsky

90,434

X is a Row reduced Echelon to A – Mathematics enjoyer Jan 11 '24 at 15:24
I am sorry I didn't know how to translate that to English so I skipped it thinking it is not important – Mathematics enjoyer Jan 11 '24 at 15:25
@Mathematicsenjoyer It seems this additional requirement is not a problem for a counterexample, because we can swap $X$ with $A$ and swap $Y$ with $B$, right? – Alex Ravsky Jan 11 '24 at 19:08
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Oh you are correct – Mathematics enjoyer Jan 11 '24 at 19:33

If $A$ can be row reduced Echelon to $X$ and $B$ can be row reduced Echelon to $Y$ Is it true that that $AB$ can be row reduced to $XY$?

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