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Question #41140

Difficulty: 8/10 | 🟢 Free

The Question

Let $M = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}$ and $S = \{ aM + bM^2 \mid a, b \in \mathbb{R} \}$ . A binary operation $*$ on $S$ is defined by $A * B = A + B + c AB$ , where $c$ is a non-zero real constant. For any $A \in S$ , let $A^{\circ n} = A * A * \dots * A$ ( $n$ times). If $X \in S$ is the inverse of $M^{\circ 5}$ under the operation $*$ , then $X$ is equal to:

Options

A.
$-5M + 15c M^2$
B.
$-5M + 10c M^2$
C.
$-5M - 15c M^2$
D.
$5M - 10c M^2$

Check Your Answer

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

Binary Operation Closure|Inverse Existence Criterion|Nilpotent Matrix Logic

Why This Question Is Tricky

Common Mistakes

This question tests the group structure of singular matrices under a custom binary operation.

The primary trap is a 'Logical Deadlock': students typically associate the existence of an inverse with non-zero determinants.

Here, every matrix in $S$ is nilpotent (and thus singular with det=0), yet the set forms a closed group under $*$ .

To solve it efficiently, students must avoid brute-force algebra (which leads to massive calculation errors for $n=5$ ) and recognize the isomorphism $A \mapsto I+cA$ , transforming the operation into standard multiplication.

They must then apply the binomial expansion for negative indices, using the nilpotent property $M^3=O$ to naturally terminate the infinite series.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Binary Operation Closure
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free