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Question #41234

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = I + K$ be a $4 \times 4$ matrix, where $I$ is the identity matrix and $K = [k_{ij}]$ is a skew-symmetric matrix with real entries. It is given that $k_{12}=1$ , $k_{13}=2$ , $k_{14}=3$ , $k_{24}=2$ , $k_{34}=1$ , and $k_{23}=-x$ for some $x > 0$ . If $\det(K) = 36$ , find the value of $\det(A)$ .

Options

A.
37
B.
57
C.
65
D.
17

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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|Inversion and Determinant Product|Row/Column Exchange Parity

Why This Question Is Tricky

Common Mistakes

This question targets the non-linear properties of determinants and the spectral structure of skew-symmetric matrices.

The primary trap is the false linearity assumption $\det(I+K) = \det(I) + \det(K) = 37$ .

A secondary trap lies in the Pfaffian expansion of a $4 \times 4$ skew-symmetric matrix: students frequently miss the negative sign in the parity of permutations ( $k_{12}k_{34} - k_{13}k_{24} + k_{14}k_{23}$ ), leading to an incorrect value of $x$ .

Finally, by recognizing that $\det(I+K)$ is simply the characteristic polynomial $P(\lambda) = \det(\lambda I - K)$ evaluated at $\lambda = -1$ , the student bypasses a brutal and error-prone $4 \times 4$ determinant expansion, rewarding deep conceptual understanding of matrix invariants.

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