Consider the following relation R on the set of realsquare matrices of order 3. `R = {(A, B)| A = P^-1 BP` for some invertible matrix P} Statement `I` R is an equivalence relation. Statement `II` For any two invertible `3xx3` matrices `M and N`, `(MN)^-1 = N^-1 M^-1`

Options

A. Statement-1 is True, Statement-2 is true, Statement-2 is a correct explanation for statement-1.
B. Statement-1 is True, Statement-2 is true, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True.

Answer

A

Explanation

Reflexivity of R: For any square matrix A of order 3,
we have
`A=I^(-1)Al`. Where I is invertible
`implies (A, A) in R`
So, R is reflexive.
Symmetry of R: Let (A, B) `in R`. Then,
`A = P^(-1)` BP for some invertible matrix P
`implies B=PAP^(-1)=(P^(-1))^(-1)AP^(-1)` for some invertible matrix `P^(-1)`
`implies (B, A)in R`
So, R is symmetric.
Transitivity of R: Let `(A, B) in R` and `(B, C) in R`. Then,
`A=P^(-1)BP` and `B=Q^(-1)CQ` for some invertible matrices P and Q
implies `A=P^(-1)(Q^(-1)CQ)P`
`implies A=(QP)^(-1)C(QP)`
`implies(A,C) in R`
implies R is transitive.
Hence, R is an equivalence relation.
Consequently, statement-1 is correct.
Clearly, statement-2 is also true.
Also, we have used this statement for proving the correctness of statement-1.
Hence, statement-2 is a correct explanation for statement-1.