Correct option is (3) R is an equivalence relation
A and B are matrices of n × n order & ARB if there exists a non singular matrix P(det(P) ≠ 0) such that PAP–1 = B
For reflexive
ARA ⇒ PAP–1 = A ...(1) must be true for P = I, Eq.(1) is true so 'R' is reflexive
For symmetric
ARB ⇔ PAP–1 = B ...(1) is true
for BRA if PBP–1 = A ...(2) must be true
\(\because\) PAP–1 = B
P–1PAP–1 = P–1B
IAP–1P = P–1BP
A = P–1BP ...(3)
from (2) & (3) PBP–1 = P–1BP
can be true some P = P–1 ⇒ P2 = I (det(P) ≠ 0)
So 'R' is symmetric
For transitive
ARB ⇔ PAP–1 = B... is true
BRC ⇔ PBP–1 = C... is true
now PPAP–1P–1 = C
P2A(P2)–1 = C
⇒ ARC So 'R' is transitive relation
⇒ Hence R is equivalence.