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Question #40489

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a $3 \times 3$ matrix with complex entries such that $\text{tr}(A) = 4 - i$ and $\text{tr}(A^2) = 5 + 2i$ . Let $\alpha$ and $\beta$ be the roots of the quadratic equation $z^2 - \text{tr}(A)z + \text{tr}(\text{adj}(A)) = 0$ Find the value of $|\alpha|^2 + |\beta|^2$ .

Options

A.
15
B.
17
C.
25
D.
29

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

Quadratic Equations with Complex Coefficients|Matrix Trace Properties|Adjoint-Determinant Relation

Why This Question Is Tricky

Common Mistakes

This problem tests the intersection of matrix invariants and complex quadratic equations.

The primary trap lies in the assumption that $|\alpha|^2 + |\beta|^2$ can be evaluated using symmetric sum formulas like $|\alpha+\beta|^2 - 2|\alpha\beta|$ , which is invalid for complex numbers since $|\alpha|^2 + |\beta|^2 \neq |\alpha+\beta|^2 - 2\text{Re}(\alpha\bar{\beta})$ .

This forces the student to explicitly solve the quadratic equation.

To even form the equation, students must know the $3 \times 3$ matrix identity $\text{tr}(\text{adj}(A)) = \frac{1}{2}((\text{tr}(A))^2 - \text{tr}(A^2))$ .

A secondary trap is the complex discriminant $\Delta = -5 + 12i$ ; students who incorrectly expand $(4-i)^2$ as $17-8i$ (ignoring $i^2 = -1$ ) will face a non-perfect square discriminant, leading to a dead end.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Quadratic Equations with Complex Coefficients
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free