We are given a relation $R$ on the class of all $n\times n$ real matrices, defined by

This is the usual similarity relation on matrices. We must check whether it is reflexive, symmetric, and transitive.

1. Reflexive

We need to check whether every matrix $A$ satisfies $A R A$.

Take

where $I$ is the identity matrix. Since $I$ is non-singular,

Hence,

So $R$ is reflexive.

2. Symmetric

Assume

Then there exists a non-singular matrix $P$ such that

We must show that $B R A$.

Since $P$ is non-singular, $P^{-1}$ also exists and is non-singular. Rewrite:

Multiply suitably to solve for $A$:

Now let

Then $Q$ is non-singular, and

Thus,

So $R$ is symmetric.

3. Transitive

Assume

Then there exist non-singular matrices $P$ and $Q$ such that

and

Substitute $B = PAP^{-1}$ into the second equation:

Rearrange:

Since $P$ and $Q$ are non-singular, $QP$ is also non-singular. Therefore,

So $R$ is transitive.

4. Conclusion

The relation $R$ is:

• reflexive,
• symmetric,
• transitive.

Hence $R$ is an equivalence relation.

Therefore, the correct option is

5. Comparison with stored correct answer

Stored correct answer: $D$

Our derived answer: $D$

They match.