Relations
Equivalence Relations
JEE Main 2021
Grade 11
Question:
Define $A R B$ on $n \times n$ real matrices iff there exists a non-singular $P$ such that $PAP^{-1} = B$. Then $R$ is:
(1) reflexive only
(2) an equivalence relation
(3) symmetric but not transitive
(4) reflexive and symmetric but not transitive
Step-by-Step Solution
Key Concept: This is matrix similarity. Reflexive: $A = IAI^{-1}\checkmark$. Symmetric: if $PAP^{-1} = B$ then $A = P^{-1}B(P^{-1})^{-1}\checkmark$. Transitive: if $A \sim B$ and $B \sim C$, then $A \sim C$ (compose the non-singular matrices)$\checkmark$. Equivalence.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)