JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Sets Relations and Functions

Taking $P=I$ gives $IA{I}^{-1}=A$, so each matrix is related to itself and $R$ is reflexive. If $PA{P}^{-1}=B$, then multiplying on the left by $P^{-1}$ and on the right by $P$ gives $P^{-1}B{\left(P^{-1}\right)}^{-1}=A$, and $P^{-1}$ is nonsingular, so $R$ is symmetric. If $P_{1}A{P}_1^{-1}=B$ and $P_{2}B{P}_2^{-1}=C$, then

and $P_2{P}_1$ is nonsingular as a product of nonsingular matrices, so $R$ is transitive. Being reflexive, symmetric and transitive, $R$ is an equivalence relation.