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Question #68093

Difficulty: 8/10 | 🟢 Free

The Question

Let $M$ be a $3 \times 3$ matrix with complex entries. Let the eigenvalues of $M$ be $z_1, z_2, z_3$ such that $|z_1| = |z_2| = |z_3| = \omega$ , where $\omega > 0$ . If $\text{Tr}(M) = \alpha$ and $\text{Tr}(\text{adj}(M)) = \beta$ satisfy $|\beta| = 3\omega^2$ , which of the following represents $\det(M)$ ?

Options

A.
$\frac{\alpha^3}{27}$
B.
$\frac{\alpha |\alpha|^2}{9}$
C.
$\frac{|\alpha|^3}{27}$
D.
$\frac{\alpha^3}{9}$

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

Matrix Trace Properties|Vieta’s Formulas in Complex Domain|Triangle Inequality Mind Games

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 intersection of linear algebra (matrix invariants) and complex analysis.

The 'Lethal Collision' occurs when students must relate the trace of the adjugate matrix to the conjugate of the trace using the property $\bar{z} = \omega^2/z$ .

The ultimate trap lies in the magnitude condition $|\alpha| = 3\omega$ : students often fail to recognize that reaching the upper bound of the complex triangle inequality ( $|z_1+z_2+z_3| = |z_1|+|z_2|+|z_3|$ ) strictly forces the eigenvalues to be identical.

Without this geometric realization, the algebraic system appears underdetermined, penalizing those who attempt to solve it purely through polynomial manipulation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Matrix Trace Properties
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free