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Question #67144

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be an $n \times n$ Hermitian matrix (where $n$ is an odd integer, $n \ge 3$ ) such that $\det(A) < 0$ . Let $z$ be a non-zero complex number with principal argument $\theta$ satisfying $\frac{\pi}{n} < \theta < \frac{3\pi}{2n}$ . If $M = z^2 A$ , which of the following represents the principal argument of $\det(M)$ ?

Options

A.
$2n\theta - 3\pi$
B.
$2n\theta - \pi$
C.
$2n\theta - 2\pi$
D.
$2n\theta + \pi$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Determinant Scaling Property|Euler's Form and Rotations|The Modulus Trap

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the 'Modulus Trap' combined with the determinant scaling property.

Students often correctly apply $\det(kA) = k^n \det(A)$ but fail to account for the negative sign of the Hermitian matrix's determinant, which adds a $\pi$ phase shift.

Furthermore, the trap forces students to carefully evaluate the bounds of the raw argument.

A common mistake is subtracting only $2\pi$ to normalize the argument, which leaves it in $(\pi, 2\pi)$ , outside the strict principal range of $(-\pi, \pi]$ .

Correctly identifying that $4\pi$ must be subtracted is the core discriminator.

The condition that $n$ is odd acts as a distractor, tempting students to incorrectly apply $(-1)^n$ to the scaling factor $z^2$ .

Quick Facts About This Question

Question Information

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