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I am trying to write a proof for following statement:

Let A be an n×n matrix and let Aˆ be a matrix that is obtained from A by scrambling the rows. Show that there is a unique n×n permutation matrix P such that Aˆ = PA

So, I was thinking to take a matrix (let say 3 by 3 ) and show explicitly that among three possible combinations of P only one satisfies the condition A^ = PA, i.e. just to show the calculations and say that all results are different and there is only one where A^ = PA. Is there any other more general way to show the uniqueness of that permutation matrix P?

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Hint Look at $$A=\begin{bmatrix} 1&1&1&..1 \\ 1&1&1&..1 \\ 1&1&1&..1 \\ ...&...&...&...\\ 1&1&1&..1 \\ \end{bmatrix}$$

or even $A=0_n$.

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  • $\begingroup$ Hi @N.S. could you explain little more detailed, because I don't really see what we can get out of this 1 and 0 matrices $\endgroup$ – ViniLL Oct 7 '18 at 18:12
  • $\begingroup$ @DanF. If $P$ is ANY permutation matrix, what is $PA$? $\endgroup$ – N. S. Oct 7 '18 at 18:52
  • $\begingroup$ It is just a matrix with rows switched depending on the P $\endgroup$ – ViniLL Oct 7 '18 at 20:09
  • $\begingroup$ @DanF. And what do you get if you switch the rows of a matrix will ALL entries equal? $\endgroup$ – N. S. Oct 7 '18 at 23:26
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    $\begingroup$ @DanF. And again: If you manage to prove it, for this particular matrix you prove something which is not true... So let me ask it clearly: how can you prove something which is not true? $\endgroup$ – N. S. Oct 8 '18 at 20:14

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