Question

In the set of all $3\times 3$ real matrices a relation is defined as follows. A matrix $A$ is related to a matrix $B$, if and only there is a non-singular $3\times 3$ matrix $P$, such that $B=P^{-1}AP$. This relation is

• A. reflexive,symmetric but not transitive
• B. reflexive, transitive but not symmetric
• C. symmetric, transitive but not reflexive
• D. an equivalence relation

Answer 1

The correct option is D an equivalence relation

Let the relation defined as
$R=(A,B)|B=P^{-1}AP$
For reflexive $A=I^{-1}AI$
$\Rightarrow$ $(A,A)\in R$
$\Rightarrow$ $R$ is reflexive
For symmetric:

Let $(A,B)\in R$
$\because$ $B=P^{-1}AP$
$\Rightarrow$ $PB=AP$ $\Rightarrow$ $PB{P}^{-1}=A$
$\Rightarrow$ $(B,A)\in R$ $\Rightarrow$ is symmetric
For transitive:
Let $(A,B)\in R,(B,C)\in R$
$\because$ $A=P^{-1}BP$ and $B=Q^{-1}CQ$
$\Rightarrow$ $A=P^{-1}{Q}^{-1}CAP={\left(QP\right)}^{-1}C\left(QP\right)$
$\Rightarrow$ $(A,C)\in R$
$\Rightarrow$ $R$ is transitive
Since, $R$ is reflexive, symmetric and transitive
So, $R$ is an equivalence relation.