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?