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Question #41236

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = (\lambda^2 - 1)I_3$ and $K$ be a $3 \times 3$ skew-symmetric matrix with $K_{12} = K_{13} = K_{23} = 1$ . Let $A = S + K$ and $B = \begin{pmatrix} \lambda^2 + \lambda - 2 \\ \lambda^2 + |\lambda| - 2 \\ \lambda - |\lambda| \end{pmatrix}$ . If $S_1$ is the set of all real values of $\lambda$ for which the system of equations $AX = B$ has infinitely many solutions, and $S_2$ is the set of all real values of $\lambda$ for which the system has no solution, then find $S_1$ and $S_2$ .

Options

A.
$S_1 = \{1\}, S_2 = \{-1\}$
B.
$S_1 = \{-1\}, S_2 = \{1\}$
C.
$S_1 = \emptyset, S_2 = \{1, -1\}$
D.
$S_1 = \{1, -1\}, S_2 = \emptyset$

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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

Symmetric and Skew-Symmetric Decomposition|System of Linear Equations (Cramer/Inverse)|The Modulus Trap

Why This Question Is Tricky

Common Mistakes

The problem tests the consistency of a linear system AX=B where A is decomposed into symmetric and skew-symmetric parts.

The core trap is the 'Modulus Trap' combined with the properties of skew-symmetric matrices.

Students often attempt to find the inverse directly or miscalculate the determinant by ignoring the +3x term from the skew-symmetric cross-terms, missing that A is singular only when the symmetric part vanishes (x=0).

Furthermore, even if they find the singular points, the modulus functions in vector B create a secondary trap: students must derive the exact row-dependence condition to check consistency, revealing that the modulus condition selectively makes one root consistent and the other inconsistent.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Symmetric and Skew-Symmetric Decomposition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free