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If $rref(A)$ = $I$ and $rref(B)$ = $I$, can we say that $A$ and $B$ are row equivalent?

My intuition says Yes, but I can't really prove it. If this is true, does that mean that all invertible matrices of the same dimensions are row equivalent?

(Assume that $A$, $B$ and $I$ are $n$ by $n$)

I couldn't find an answer to this, maybe it's there somewhere, but in that case I couldn't understand it.

Thanks.

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Yes -- if you can show that the inverse of elementary row operations are also elementary row operations, the proof is essentially already done (just apply the inverse of matrix B's operations to both sides).

And yes, this fact (plus a proof that an invertible matrix's row-reduced echelon form is always the identity matrix) does imply that all invertible matrices of the same dimension are row-equivalent.

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  • $\begingroup$ Ok, thanks! It's nice to get your intuition confirmed! $\endgroup$ – Morgan May 7 '15 at 19:53

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