Permute columns by pre-multiplying and rows by post-multiplying?

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If such $\bm P'$ works for all matrices, then for all $\bm A$, $$ \bm {AP} = \bm {P'A}, $$ then specifically it works for the identity matrix $\bm I$, i.e. $$ \bm {IP} = \bm {P'I}, $$ then the only candidate of $\bm P'$ is $\bm P$ again. But clearly $$ \bm {PA} = \bm {AP} $$ only holds for specific matrices $\bm A$.

Conclusion: maybe for some $\bm A$, there exists $\bm P'$ that $\bm {P'A}$ swap two columns of $\bm A$, but there exists no universal matrices $\bm P'$.