Currently, I'm trying to understand the idea of matrix similarity. As a toy example, I am thinking about $Id$ and $-Id$. Now, I do not think that these matrices are similar, and here is my proposed reasoning:
Assume that there exists an invertible matrix $P$ such that $Id = P^{-1}(-Id)P$. Since $-Id = (-1)(Id)$, we see that $Id = P^{-1}(-1)(Id)P = (-1)P^{-1}(Id)P = (-1)P^{-1}P = (-1)Id = -Id$ which is a contradiction. Therefore, $Id$ and $-Id$ are not similar.
(NOTE: I could have also done it this way, I'm pretty sure at least: Assume that there exists an invertible matrix $P$ such that $-Id = P^{-1}(Id)P$. Since $P^{-1}(Id)P = P^{-1}P = Id$, we see that $-Id = P^{-1}(Id)P = P^{-1}P = Id$ is a contradiction, and so therefore $-Id$ and $Id$ are not similar.)
Is this reasoning valid? Thanks in advance: it greatly helps to clarify my thinking process with the help of generous and insightful users on here!
Yes, correct, $\rm id$ and $-\rm id$ are similar only if $1=-1$, i.e. over a field of characteristic $2$.