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Question #40540

Difficulty: 8/10 | 🟢 Free

The Question

Let $M$ be a $2 \times 2$ matrix given by $M = \begin{pmatrix} |x^2 - 4x - 2| - 2 & -1 \\ 1 & |x-2| - 4 \end{pmatrix}$ . If there exists a $2 \times 2$ matrix $N$ with real entries such that $M N - N M = M$ , find the sum of all possible real values of $x$ .

Options

A.
4
B.
10
C.
12
D.
0

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Modulus Inequality Logic|System of Linear Equations (Cramer/Inverse)|Matrix Multiplication Non-Commutativity

Why This Question Is Tricky

Common Mistakes

This question tests the properties of the matrix commutator $[M, N] = M$ combined with modulus piecewise logic.

A common trap is assuming that taking the trace ( $\text{tr}(M) = 0$ ) is the only necessary condition, leading students to solve $|x^2 - 4x - 2| + |x-2| - 6 = 0$ and sum all its roots (yielding the trap answer 10).

However, matrix non-commutativity in this specific Lie algebra relation forces $M$ to be strictly nilpotent, meaning $\det(M) = 0$ must also hold.

The modulus definitions act as a filter, where only the intersection of the trace and determinant conditions yields the true solution.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Modulus Inequality Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free