Do elementary row operations give a similar matrix transformation? So we define two matrices $A,B$ to be similar if there exists an invertible square matrix $P$ such that $AP=PB$. I was wondering if $A,B$ are related via elementary row operations (say, they are connected via some permutation rows for example) then are the necessarily similar?
Obviously swapping rows multiplies the determinant by $-1$ but I was thinking if we permute rows in pairs, would this allow us to construct a similarity transformation?

Answer is NO. Simplest exmple is as follows.
Consider two satrices  $ \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix} $ and $ \begin{pmatrix} 1 & 1 \\ 0 & 1 \\ \end{pmatrix} $. Both are row equivalent. But not similar.
