linear algebra - 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

Every invertible matrix is equivalent via row operations to the identity matrix, and the identity matrix is only similar to itself.

This also gives a counterexample to the permutation question; the identity matrix is not similar to a non-identity permutation matrix.