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Question #41144

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all $2 \times 2$ matrices of the form $M(x, y) = \begin{pmatrix} x & y \\ 0 & \frac{1}{x} \end{pmatrix}$ , where $x \in \mathbb{R} \setminus \{0\}$ and $y \in \mathbb{R}$ . A binary operation $\star$ is defined on $S$ by $A \star B = A B A^{-1}$ for all $A, B \in S$ . Let $L$ be the set of all left identities and $R$ be the set of all right identities for the operation $\star$ in $S$ . Which of the following statements is correct?

Options

A.
$L = \{I, -I\}$ and $R = \emptyset$
B.
$L = \{I\}$ and $R = \emptyset$
C.
$L = \emptyset$ and $R = \{I, -I\}$
D.
$L = \{I, -I\}$ and $R = \{I, -I\}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Identity Element Uniqueness|Matrix Multiplication Non-Commutativity|Triangular/Diagonal Matrix Shortcut

Why This Question Is Tricky

Common Mistakes

The problem defines a non-standard binary operation (conjugation) on a specific group of matrices.

The primary trap is the assumption that the identity element must be unique and equal to the standard identity matrix $I$ .

Students often prematurely apply properties of standard matrix multiplication or assume that if $I$ works partially, it is the unique two-sided identity.

By forcing the calculation of left and right identities separately, the question reveals that there are exactly two left identities ( $I$ and $-I$ ) and zero right identities, neutralizing rote memorization of group axioms.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Identity Element Uniqueness
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free