Question

Define a relation R over a class of n × n real matrices A and B as "ARB if there exists a non-singular matrix P such that PAP^–1 = B".

Then which of the following is true ?

(1) R is symmetric, transitive but not reflexive,

(2) R is reflexive, symmetric but not transitive 

(3) R is an equivalence relation

(4) R is reflexive, transitive but not symmetric

Answer 1

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.