Let $A$ be a nonzero $3 \times 3$ matrix with entries in $\mathbb{R}$. For any non-empty subset $\mathcal{S}$ of the set of all $3 \times 3$ matrices with entries in $\mathbb{R}$, we define a relation $\alpha_A$ on $\mathcal{S}$ by setting $X \alpha_A Y$ if $X A=A Y$. Then $\alpha_A$ is an equivalence relation on $\mathcal{S}$

• A

  for any subset $\mathcal{S}$
• B

  if all matrices in $\mathcal{S}$ are symmetric
• Correct answer
  C

  if it is reflexive
• D

  only if all elements of $\mathcal{S}$ are diagonal matrices

Hence, the correct answer is option: 'C'.