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 ?
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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.
$R$ is reflexive and symmetric, but not transitive.
$R$ is reflexive, but neither symmetric nor transitive.
$R$ is an equivalence relation.
$R$ is symmetric and transitive, but not reflexive.
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The Correct Option isA
Solution and Explanation
Step 1: Understanding the Question
We are given a relation \( R \) defined on the set of all \( 3 \times 3 \) real matrices \( M \).
Two matrices \( A \) and \( B \) are related, i.e., \( (A, B) \in R \), if and only if the determinant of their difference \( \det(A - B) \) is an integer.
We need to check the three properties of relations: reflexivity, symmetry, and transitivity.
Step 2: Key Formula or Approach
Reflexivity: \( (A, A) \in R \) for all \( A \in M \).
Symmetry: \( (A, B) \in R \implies (B, A) \in R \).
Transitivity: \( (A, B) \in R \) and \( (B, C) \in R \implies (A, C) \in R \).
We use the determinant property: \( \det(-X) = (-1)^n \det(X) \), where \( n \) is the order of the matrix. Here \( n = 3 \).
Step 3: Detailed Explanation
Reflexivity:
\( \det(A - A) = \det(0) = 0 \). Since 0 is an integer, \( (A, A) \in R \). Hence, \( R \) is reflexive.
Symmetry:
Suppose \( (A, B) \in R \), i.e., \( \det(A - B) = k \), \( k \in \mathbb{Z} \).
Then:
\( \det(B - A) = \det(-(A - B)) = (-1)^3 \det(A - B) = -k \in \mathbb{Z} \).
Hence, \( (B, A) \in R \). So \( R \) is symmetric.
Transitivity:
Suppose \( (A, B) \in R \) and \( (B, C) \in R \). That is:
\( \det(A - B) \in \mathbb{Z}, \quad \det(B - C) \in \mathbb{Z} \).
Consider a counterexample:
\( A - B = I = \begin{pmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{pmatrix}, \quad \det(A - B) = 1 \in \mathbb{Z} \)
\( B - C = \begin{pmatrix}1/2 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{pmatrix}, \quad \det(B - C) = 0 \in \mathbb{Z} \)
Then:
\( A - C = (A - B) + (B - C) = \begin{pmatrix}3/2 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{pmatrix}, \quad \det(A - C) = \frac{3}{2} \cdot 1 \cdot 1 = 1.5 \)
Since 1.5 is not an integer, \( (A, C) \notin R \). Hence, \( R \) is not transitive.
Step 4: Final Answer
The relation \( R \) is reflexive and symmetric, but not transitive.
