Question

Let $M$ be the set of all $3\times 3$ matrices with real entries. Consider the relation $R$ on $M$ given by $R = \{(A, B) \in M \times M : \det(A-B) \text{ is an integer}\}$. Which one of the following statements is Correct ?

• A. $R$ is reflexive and symmetric, but not transitive.
• B. $R$ is reflexive, but neither symmetric nor transitive.
• C. $R$ is an equivalence relation.
• D. $R$ is symmetric and transitive, but not reflexive.

Hint:

A simple diagonal matrix addition counterexample is the easiest way to test transitivity of determinant relations.
Since \(\det(X+Y) \neq \det(X) + \det(Y)\). in general, transitivity rarely holds for determinant-based conditions.

Answer 1

The Correct Option is A

The question asks us to analyze a relation \( R \) on the set of all \( 3 \times 3 \) matrices with real entries. The relation is defined as \( R = \{(A, B) \in M \times M : \det(A-B) \text{ is an integer}\} \). We need to determine the properties of this relation. 

1. Check for Reflexivity: A relation \( R \) is said to be reflexive if every element is related to itself.
   • Consider a matrix \( A \in M \). We need to check if \( (A, A) \in R \).
   • Since \( \det(A-A) = \det(\mathbf{0}) = 0 \), and 0 is an integer, \( (A, A) \in R \).
   • This implies that \( R \) is reflexive.
2. Check for Symmetry: A relation \( R \) is symmetric if for any \( (A, B) \in R \), \( (B, A) \in R \) as well.
   • Assume \( (A, B) \in R \), which means \( \det(A-B) \) is an integer.
   • We need to check if \( (B, A) \in R \), meaning that \( \det(B-A) \) should also be an integer.
   • Note that \( \det(A-B) = -\det(B-A) \). Since the negative of an integer is also an integer, \( (B, A) \in R \).
   • Thus, \( R \) is symmetric.
3. Check for Transitivity: A relation is transitive if whenever \( (A, B) \in R \) and \( (B, C) \in R \), then \( (A, C) \in R \).
   • Assume \( (A, B) \in R \) and \( (B, C) \in R \), meaning \( \det(A-B) \) and \( \det(B-C) \) are integers.
   • We need to determine if \( \det(A-C) \) is also an integer.
   • However, knowing that the determinant of the difference of each pair is an integer does not ensure \( \det(A-C) \) is an integer without additional assumptions. Thus, transitivity might not hold in all cases.

Based on the above analysis, the correct statement is: $R$ is reflexive and symmetric, but not transitive.