Define a relation R over a class of n × n real matrices A and B as " A R B iff there exists a non-singular matrix P such that P A P - 1 = B ". Then which of the following is true ?

Step-by-step solution

Given A and B are matrices of n × n order and A R B iff there exists a non-singular matrix P   det P ≠ 0 such that P A P - 1 = B . Reflexivity Check : A R A ⇒ P A P - 1 = A which is true for P = I . So, R is reflexive relation. Symmetric Check : A R B ⇒ P A P - 1 = B ⇒ P - 1 P A P - 1 P = P - 1 B P ⇒ I A I = P - 1 B P ⇒ P - 1 B P = A ⇒ B R A for matrix P - 1 . So, R is symmetric relation. Transitivity Check : A R B ⇒ P A P - 1 = B and B R C ⇒ P B P - 1