Sets, Relations and Function: Define a relation R over a class of ntimes n real matrices A and B… | JEE Main 2021

JEE Main 2021 (Online) 18th March Evening ShiftQuestion 1

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

Sets, Relations and Function

$R$ is reflexive, symmetric but not transitive$R$ is reflexive, transitive but not symmetric$R$ is symmetric, transitive but not reflexive,$R$ is an equivalence relation✓ Correct answer

Solution

Reflexive ($P=I$), symmetric ($B=PAP^{-1}\Rightarrow A=P^{-1}BP$), transitive (product of invertibles). So similarity is an equivalence relation. Option (D).

Sets, Relations and Function - JEE Mathematics question (JEE Main 2021 (Online) 18th March Evening Shift)

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