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In the set of all $3 \times 3$ real matrices a relation is defined as follows. A matrix $A$ is related to a matrix $B,$ if and only if there is a non-singular $3 \times 3$ matrix $P,$ such that $B=P^{-1} A P .$ This relation is
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an equivalence relation
Let the relation defined as $R=\left\{(A, B) \mid B=P^{-1} A P\right\}$
For reflexive, $A=I^{-1} A l$
$\Rightarrow (A, A) \in R$
$\Rightarrow R$ is reflexive
For symmetric $\operatorname{Let}(A B) \in R$
$\because$
$B=P^{-1} A P$
$\Rightarrow P B=A P \Rightarrow P B P^{-1}=A$
$\Rightarrow$
$A=\left(P^{-1}\right)^{-1} B\left(P^{-1}\right)$
$\Rightarrow (B, A) \in R \Rightarrow R$ is symmetric.
For transitive
Let $(A, B) \in R,(B, C) \in R$
$\because A=P^{-1} B P$ and $B=Q^{-} C Q$
$\Rightarrow A=P^{-} Q^{-1} C Q P=(Q P)^{-1} C(Q P)$
$\Rightarrow (A, C) \in R$
$\Rightarrow R$ is transitive. since, $A$ is reflexive, symmetric and transitive. So, $R$ is an equivalence relation.
For reflexive, $A=I^{-1} A l$
$\Rightarrow (A, A) \in R$
$\Rightarrow R$ is reflexive
For symmetric $\operatorname{Let}(A B) \in R$
$\because$
$B=P^{-1} A P$
$\Rightarrow P B=A P \Rightarrow P B P^{-1}=A$
$\Rightarrow$
$A=\left(P^{-1}\right)^{-1} B\left(P^{-1}\right)$
$\Rightarrow (B, A) \in R \Rightarrow R$ is symmetric.
For transitive
Let $(A, B) \in R,(B, C) \in R$
$\because A=P^{-1} B P$ and $B=Q^{-} C Q$
$\Rightarrow A=P^{-} Q^{-1} C Q P=(Q P)^{-1} C(Q P)$
$\Rightarrow (A, C) \in R$
$\Rightarrow R$ is transitive. since, $A$ is reflexive, symmetric and transitive. So, $R$ is an equivalence relation.
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