Consider the following relation R on the set of realsquare matrices of order 3. `R = {(A, B)| A = P^-1 BP` for some invertible matrix P} Statement `I` R is an equivalence relation. Statement `II` For any two invertible `3xx3` matrices `M and N`, `(MN)^-1 = N^-1 M^-1`

Options

A. Statement -1 is true, Statement -2 is true and Statement -2 is correct explanation for Statement -1
B. Statement -1 is true, Statement -2 is true and Statement -2 is not correct explanation for Statement -1
C. Statement -1 is true, Statement -2 is false
D. Statement -1 is false Statement -2 is true

Explanation

To solve the given problem, we need to analyze the two statements regarding the relation \( R \) on the set of real square matrices of order 3 and the properties of invertible matrices.

### Step 1: Analyze Statement I

**Statement I**: \( R \) is an equivalence relation.

To determine if \( R \) is an equivalence relation, we need to check three properties: reflexivity, symmetry, and transitivity.

1. **Reflexivity**: A relation \( R \) is reflexive if for every matrix \( A \), \( (A, A) \in R \).
- For \( A \) to be related to itself, we need \( A = P^{-1} A P \) for some invertible matrix \( P \).
- If we take \( P = I \) (the identity matrix), then \( A = I^{-1} A I = A \).
- Therefore, \( R \) is reflexive.

2. **Symmetry**: A relation \( R \) is symmetric if whenever \( (A, B) \in R \), then \( (B, A) \in R \).
- Assume \( (A, B) \in R \), meaning \( A = P^{-1} B P \) for some invertible matrix \( P \).
- To show symmetry, we need to express \( B \) in terms of \( A \).
- From \( A = P^{-1} B P \), we can rearrange to find \( B \):
\[ B = P A P^{-1} \]
- For symmetry, we need \( B = Q^{-1} A Q \) for some invertible matrix \( Q \). However, we cannot guarantee that \( Q \) can be chosen such that this holds for all \( A \) and \( B \). Thus, \( R \) is not symmetric.

Since \( R \) is not symmetric, it cannot be an equivalence relation.

### Conclusion for Statement I:
Statement I is **false**.

### Step 2: Analyze Statement II

**Statement II**: For any two invertible \( 3 \times 3 \) matrices \( M \) and \( N \), \( (MN)^{-1} = N^{-1} M^{-1} \).

To verify this statement, we can use the property of inverses in matrix multiplication:
- The inverse of a product of two matrices is given by:
\[ (MN)^{-1} = N^{-1} M^{-1} \]
This property holds for any two invertible matrices, regardless of their order.

### Conclusion for Statement II:
Statement II is **true**.

### Final Answer:
- Statement I is false.
- Statement II is true.

### Summary:
The correct option is that Statement I is false and Statement II is true.

---