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 $3\times 3$ 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
$\left(A,A\right)\in R$
$\Rightarrow A=P^{-1}AP$
which $.......$ for $P=I$
$\therefore$ refrexive
for symmetry
As $\left(A,B\right)\in$ R for matrix P
$A=P^{-1}BP$
$\Rightarrow PA{P}^{-1}=B$
and $B=P^{-1}CP$
$\Rightarrow A=P^{-1}\left(P^{-1}CP\right)P$
$\Rightarrow A={\left(P^{-1}\right)}^{2}C{P}^2$
$\Rightarrow A={\left(P^2\right)}^{-1}C\left(P^2\right)$
$\therefore \left(A,C\right)\in$ R for matrix $P^2$
$\therefore$ R is transitive
so R is equivalence