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Question #40474

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all $2 \times 2$ matrices $A = \begin{pmatrix} z_1 & z_2 \\ z_3 & z_4 \end{pmatrix}$ with complex entries such that $A^T A = I$ , where $I$ is the identity matrix. If $|z_k| = 1$ for $k=1,2,3,4$ , find the number of elements in $S$ .

Options

A.
0
B.
8
C.
16
D.
32

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

Euler's Form and Rotations|Orthogonal Matrix Conditions|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This problem tests the intersection of orthogonal matrix properties and complex geometry.

A common trap is the 'Modulus Sign Dependency': students intuitively associate $A^T A = I$ with real rotation/reflection matrices, assuming $z_1 = \cos\theta$ and $z_3 = \sin\theta$ .

However, since $|z_1|=1$ , a real $z_1$ implies $z_1 = \pm 1$ , leading to $z_1^2 + z_3^2 = 2 \neq 1$ , which incorrectly yields 0 solutions.

The key is recognizing that the orthogonality condition applies to the squares of the complex entries, forcing $z_1^2$ and $z_3^2$ to be conjugate complex roots of unity ( $e^{\pm i\pi/3}$ ) that sum to 1, thereby unlocking the finite set of valid transformation matrices.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Euler's Form and Rotations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free