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I have seem to forgot this important fact, and I am trying to prove it to myself by looking at $A$ as matrix of the elementary row operations.

Where I seem to get stuck is that if $A$ are elementary row operations on $B$ when multiplying $B$ on the left does not multiplying $B$ from the right will no act on $B$ as elementary row operations and therefore will not result $BA=I$?

gbox
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    First, these must be square matrices for that to be true, second: under the assumption that $;AB=I;$ then also $;BA=I;$ ...right? – DonAntonio Mar 10 '17 at 09:48
  • @DonAntonio Yes, sorry, I am talking about square matrices, I am assuming $AB=I$ and trying to prove $BA=I$, namely that the right inverse and the left inverse matrices are equal – gbox Mar 10 '17 at 09:52

2 Answers2

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This is an easy application of the following proposition:

Let $A$ be a square matrix. If $AX=0$ only has the zero solution, then $A$ is invertible.

So suppose $AB=I$. Then assume $BX=0$ has two solutions $Y$ and $Z$. Then $B(Y-Z)=0$ thus $A(B(Y-Z))=0$ but also $A(B(Y-Z))=(AB)(Y-Z)=Y-Z=0$. Hence $Y=Z$. So all solutions are equal. But clearly $0$ is a solution. Thus $BX=0$ only has the zero solution and $B$ is invertible by the proposition.

Once $B$ is invertible, it is a left and right inverse to $A$ which is what you needed to show.

Mathematician 42
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Assume that $AB=I$, A and B are square

$AB=I \Rightarrow (AB)^{-1}=I^{-1} = I \Rightarrow B^{-1}A^{-1}=I \Rightarrow B(B^{-1}A^{-1})A=BIA\Rightarrow (BB^{-1})(A^{-1}A)=BIA\Rightarrow II=BIA\Rightarrow I =BA$

toliveira
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    What is $A^{-1}$ and what is $B^{-1}$? How do you know they exist? That's exactly what you need to show. – Mathematician 42 Mar 10 '17 at 10:07
  • Well I commented, I think I make a valid point. You can use determinants as a way out, but I think looking at determinants for this problem is using too much. – Mathematician 42 Mar 10 '17 at 10:11
  • @Mathematician42, sorry, I asked for a comment while your were writing it. When I posted it, I read yours and removed by request for a comment. – toliveira Mar 10 '17 at 10:16