Question

We define a binary relation $\sim$ on the set of all $3\times 3$ real matrices as $A\sim B$, if and only if there exist invertible matrices $P$ and $Q$ such that $B=PA{Q}^{-1}$. The binary relation $\sim$ is

• A. Neither reflexive nor symmetric
• B. Reflexive and symmetric but not transitive
• C. Symmetric and transitive but not reflexive
• D. An equivalence relation

Answer 1

The correct option is D An equivalence relation

Let the relation defined as
$R=\{(A,B):B=PA{Q}^{-1}\}$

For reflexive
$A=IA{I}^{-1}$
$\Rightarrow (A,A)\in R$
$\Rightarrow R$ is reflexive

For symmetric:
Let $(A,B)\in R$
$\therefore B=PA{Q}^{-1}$
$\Rightarrow BQ=PA\left(Q^{-1}Q\right)$
$\Rightarrow BQ=PA$
$\Rightarrow P^{-1}\left(BQ\right)=P^{-1}P\left(A\right)$
$\Rightarrow P^{-1}BQ=A$
$\Rightarrow (B,A)\in R$
$\Rightarrow R$ is symmetric.

For transitive:
Let $(A,B)\in R,(B,C)\in R$
Then, $A=PB{Q}^{-1}$ and $B=RC{S}^{-1}$
$\Rightarrow A=PRC{S}^{-1}{Q}^{-1}$
$\Rightarrow A=\left(PR\right)C{\left(QS\right)}^{-1}$
$\Rightarrow (A,C)\in R$
$\Rightarrow R$ is transitive.

Hence, $R$ is an equivalence relation.