Question

Consider the following relation R on the set of real square matrices of order 3.
$R =$ {$(A, B)|A = P^{-1}$ BP for some invertible matrix P}.
Statement -1 :
R is equivalence relation.
Statement - 2 :
For any two invertible $3 × 3$ matrices M and N, $(MN)^{-1} = N^{-1}M^{-1}$.

• A. Statement-1 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 1

Answer: (B)

for reflexive
$\left(A , A\right) \in R$
$\Rightarrow \quad A = P^{-1} A P$
which $.......$ for $P = I$
$\therefore \quad$ refrexive
for symmetry
As $\left(A, B\right) \in$ R for matrix P
$A = P^{-1} BP$
$\Rightarrow \quad PAP^{-1} = B$
and $ \quad B = P^{-1}CP$
$\Rightarrow \quad A = P^{-1} \left(P^{-1} CP\right)P$
$\Rightarrow \quad A = \left(P^{-1}\right)^{2} CP^{2}$
$\Rightarrow \quad A = \left(P^{2}\right)^{-1} C\left(P^{2}\right)$
$\therefore \quad\left(A, C\right) \in$ R for matrix $P^{2}$
$\therefore \quad$ R is transitive
so R is equivalence