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Question #41277

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a $3 \times 3$ Skew-Hermitian matrix with complex entries such that $A^3 + 7A - 6iI = 0$ , where $I$ is the $3 \times 3$ identity matrix and $i = \sqrt{-1}$ . Which of the following is the value of $\det((adj(A))^* - A^2)$ ?

Options

A.
$55$
B.
$-55$
C.
$343$
D.
$-343$

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

Adjoint-Determinant Relationship|Characteristic Equation|Conjugate Root Theorem

Why This Question Is Tricky

Common Mistakes

This question tests the interaction between characteristic equations, adjoint properties, and complex matrices.

The 'Lethal Collision' triggers when students misapply the Conjugate Root Theorem, erroneously assuming eigenvalues must form conjugate pairs despite the complex-coefficient polynomial.

Furthermore, students frequently assume the adjoint of a Skew-Hermitian matrix maintains the same skew-symmetry ( $(adj(A))^* = -adj(A)$ ).

However, for a $3 \times 3$ matrix, $(adj(A))^* = adj(A^*) = adj(-A) = (-1)^2 adj(A) = adj(A)$ , making it Hermitian.

Failing to account for this $n-1$ degree dependence, or making a sign error when extracting the product of roots, leads directly to the trap answers $-55$ and $-343$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Adjoint-Determinant Relationship
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free