Define a relation R over a class of n $\times$ n real matrices A and B as

"ARB iff there exists a non-singular matrix P such that PAP^$-$1 = B".

Then which of the following is true?

For reflexive relation,<br/><br/> $\forall(A, A) \in R$ for matrix $P$.<br/><br/> $\Rightarrow A=P A P^{-1}$ is true for $P=1$<br/><br/> So, $R$ is reflexive relation.<br/><br/> For symmetric relation,<br/><br/> Let $(A, B) \in R$ for matrix $P$.<br/><br/> $\Rightarrow \quad A=P B P^{-1}$<br/><br/> After pre-multiply by $P^{-1}$ and post-multiply by $P$, we get<br/><br/> $P^{-1} A P=B$<br/><br/> So, $(B, A) \in R$ for matrix $P^{-1}$.<br/><br/> So, $R$ is a symmetric relation.<br/><br/> For transitive relation,<br/><br/> Let $A R B$ and $B R C$<br/><br/> So, $A=P B P^{-1}$ and $B=P C P^{-1}$<br/><br/> Now, $A=P\left(P C P^{-1}\right) P^{-1}$<br/><br/> $\Rightarrow A=(P)^2 C\left(P^{-1}\right)^2 \Rightarrow A=(P)^2 \cdot C \cdot\left(P^2\right)^{-1}$<br/><br/> $\therefore(A, C) \in R$ for matrix $P^2$.<br/><br/> $\therefore R$ is transitive relation.<br/><br/> Hence, $R$ is an equivalence relation.