How can I show that there is a unique n×n permutation matrix P such that Aˆ = PA

Question

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?

Answer 1

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 $A=0_n$.