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Question #40458

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} z & w \\ \bar{w} & -z \end{pmatrix}$ be an involutory matrix, where $z, w \in \mathbb{C}$ . Let $B = \begin{pmatrix} 1+i & 2i & w \\ 1 & 1+i & z \\ -1 & i & 1 \end{pmatrix}$ . If $B$ is a singular matrix, let $S$ be the set of all possible values of $w$ . Find the maximum value of $|w - 2 + 2i|^2$ for $w \in S$ .

Options

A.
18
B.
$\frac{26}{3}$
C.
20
D.
2

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Algebraic Manipulation of i|Idempotent and Involutory Properties|Geometry of Complex Inequalities|Elementary Row Operations

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of matrix properties and complex number geometry.

The primary trap lies in evaluating the singularity of matrix B.

Students often attempt brute-force expansion of the determinant with complex entries, leading to algebraic mess and errors.

Recognizing the elementary row operation R_1 -> R_1 - (1+i)R_2 elegantly yields w = (1+i)z.

The secondary trap is the involutory condition A^2 = I, which forces z^2 + |w|^2 = 1.

Students frequently forget that |w|^2 is strictly real, which acts as a boundary condition forcing z^2 to be real (hence z is purely real or purely imaginary).

Failing to split into these two geometric cases results in missing the maximum value of the complex distance inequality.

Quick Facts About This Question

Question Information

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