Define a relation R over a class of n times n | vedclass

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(C) relation $R$ is defined by $A R B \iff B = P A P^{-1}$ for some non-singular matrix $P$.$1$. Reflexive: For any matrix $A$,we can choose $P = I$ (

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$R$ is an equivalence relation.

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(C) relation $R$ is defined by $A R B \iff B = P A P^{-1}$ for some non-singular matrix $P$.$1$. Reflexive: For any matrix $A$,we can choose $P = I$ (the identity matrix). Then $I A I^{-1} = I A I = A$. Thus,$A R A$ holds. So,$R$ is reflexive.$2$. Symmetric: Suppose $A R B$. Then $B = P A P^{-1}$ for some non-singular matrix $P$. We can write $A = P^{-1} B P$. Let $Q = P^{-1}$. Since $P$ is non-singular,$Q$ is also non-singular. Then $A = Q B Q^{-1}$. Thus,$B R A$ holds. So,$R$ is symmetric.$3$. Transitive: Suppose $A R B$ and $B R C$. Then $B = P A P^{-1}$ and $C = Q B Q^{-1}$ for some non-singular matrices $P$ and $Q$. Substituting $B$ in the second equation,we get $C = Q (P A P^{-1}) Q^{-1} = (Q P) A (P^{-1} Q^{-1}) = (Q P) A (Q P)^{-1}$. Since the product of two non-singular matrices $Q P$ is also non-singular,$A R C$ holds. So,$R$ is transitive.Since $R$ is reflexive,symmetric,and transitive,it is an equivalence relation.

Define a relation $R$ over a class of $n \times n$ real matrices $A$ and $B$ as $A R B$ if and only if there exists a non-singular matrix $P$ such that $P A P^{-1} = B$. Then which of the following is true?

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