Q131 / 200 · WBJEE 2014 · Mathematics

We define a binary relation - on the set of all $3 \times 3$ real matrices as $A \sim B,$ if and only if there exist invertible matrices $P$ and $Q$ such that $B=P A Q^{-1} .$ The binary relation $\sim$ is

  • A
    neither reflexive nor symmetric
  • B
    reflexive and symmetric but not transitive
  • C
    symmetric and transitive but not reflexive
  • Correct answer
    D
    an equivalence relation

Solution

Let the relation defined as $R=\left\{(A, B): B=P A Q^{-1}\right\}$ For reflexive $A=I A I^{-1}$ $\Rightarrow(A, A) \in R$ $\Rightarrow R$ is reflexive For symmetric $\operatorname{Let}(A, B) \in R$ $\therefore B=P A Q^{-1}$ $\Rightarrow \quad B Q=P A\left(Q^{-1} Q\right)$ $\Rightarrow \quad B Q=P A$ $\Rightarrow \quad P^{-1}(B Q)=P^{-1} P(A)$ $\Rightarrow \quad P^{-1} B Q=A$ $\Rightarrow(B, A) \in R$ $\Rightarrow R$ is symmetric. For transitive Let $(A, B) \in R,(B, C) \in R$ Then, $\quad A=P B Q^{-1}$ and $B=R C S^{-1}$ $\Rightarrow \quad A=P R C S^{-1} Q^{-1}$ $\begin{array}{ll}\Rightarrow & A=(P R) C(Q S)^{-1} \\ \Rightarrow & (A, C) \in R\end{array}$ $\Rightarrow R$ is transitive. Hence, $R$ is an equivalence relation.

Hence, the correct answer is option: 'D'.

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