Mathematics - Sets and Relations Question with Solution | TestHub

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 :

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

So, $R$ is reflexive relation.

Symmetric Check :

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

So, $R$ is symmetric relation.

Transitivity Check :

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

So, $PPA{P}^{-1}{P}^{-1}=C$.

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

So, $R$ is a transitive relation.

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

Hence, $R$ is an equivalence relation.