JEE Main 2021 — Sets And Relations Question with Solution

For reflexive relation,

$\forall (A,A)\in R$ for matrix $P$.

$\Rightarrow A=PA{P}^{-1}$ is true for $P=1$

So, $R$ is reflexive relation.

For symmetric relation,

Let $(A,B)\in R$ for matrix $P$.

$$\Rightarrow A=PB{P}^{-1}$$

After pre-multiply by $P^{-1}$ and post-multiply by $P$, we get

$$P^{-1}AP=B$$

So, $(B,A)\in R$ for matrix $P^{-1}$.

So, $R$ is a symmetric relation.

For transitive relation,

Let $ARB$ and $BRC$

So, $A=PB{P}^{-1}$ and $B=PC{P}^{-1}$

Now, $A=P\left(PC{P}^{-1}\right)P^{-1}$

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

$\therefore (A,C)\in R$ for matrix $P^2$.

$\therefore R$ is transitive relation.

Hence, $R$ is an equivalence relation.