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?

Not sure if this adds anything to Matt Samuel's answer which was very good, but here is another take on the question. Saying that two matrices A and B differ by some elementary row operations is equivalent to saying that there exists an invertivle matrix E such that 
$$A=EB$$
However, for A and B to be similar, there must exist some invertible nxn matrix P such that
$$
A = P^{-1}BP
$$ 
Therefore, it is the case that 
$$AE^{-1} \sim B$$
In other words, if you are going to do some preliminary (invertible) operations to the matrix of a linear transformation, you need to undo them again after performing the transformation if you want to get back a similar transformation.
