Define a relation R over a class of n × n real matrices A and B as...

Given $A$ and $B$ are matrices of $n \times n$ order and $ARB$ iff there exists a non-singular matrix $P$\left(det \left(P\right) \ne 0\right) such that $PA{P}^{- 1} = B$.

Reflexivity Check :

$ARA ⇒ PA{P}^{- 1} = A$ which is true for $P = I$.

So, $R$ is reflexive relation.

Symmetric Check :

$ARB ⇒ PA{P}^{- 1} = B ⇒ {P}^{- 1}PA{P}^{- 1}P = {P}^{- 1}BP ⇒ IAI = {P}^{- 1}BP ⇒ {P}^{- 1}BP = A ⇒ BRA$ for matrix ${P}^{- 1}$.

So, $R$ is symmetric relation.

Transitivity Check :

$ARB ⇒ PA{P}^{- 1} = B$ and $BRC ⇒ PB{P}^{- 1} = C$.

So, $PPA{P}^{- 1}{P}^{- 1} = C$.

$⇒ {P}^{2}A{\left({P}^{2}\right)}^{- 1} = C ⇒ ARC$

So, $R$ is a transitive relation.

Since, $R$ is reflexive, symmetric and transitive all.

Hence, $R$ is an equivalence relation.