Let $A$ be the set of all $3\times 3$ matrices with entries from $\{0,1\}$. Let $R$ be a relation on $A$ defined by $ARB$ if and only if there exists an invertible matrix $P$ such that $B=P^{-1}AP$. Which of the following is true?

Solution

1. Let $A$ be the set of $3\times 3$ matrices and $R$ be the relation defined by $B=P^{-1}AP$ for some invertible matrix $P$.
2. Reflexivity holds because $A=I^{-1}AI$, where $I$ is the identity matrix.
3. Symmetry holds because if $B=P^{-1}AP$, then $$\begin{array}{rl}A& =PB{P}^{-1}\\ & =(P^{-1}{)}^{-1}B{P}^{-1}\end{array}$$, implying $BRA$.
4. Transitivity holds because if $B=P^{-1}AP$ and $C=Q^{-1}BQ$, then $$\begin{array}{rl}C& =Q^{-1}(P^{-1}AP)Q\\ & =(PQ{)}^{-1}A(PQ)\end{array}$$, implying $ARC$.
5. Since the relation is reflexive, symmetric, and transitive, it is an equivalence relation.
6. Hence the answer is (A).