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Question #41210

Difficulty: 8/10 | 🟢 Free

The Question

Let $P$ be a $3 \times 3$ matrix with complex entries satisfying $P P^\theta = 2I$ and $\det(P) = 2\sqrt{2} i$ , where $P^\theta$ is the conjugate transpose of $P$ . Let $Q$ be a $3 \times 3$ matrix given by $Q = \begin{pmatrix} e^{i\pi/3} & 0 & 0 \\ 0 & e^{-i\pi/3} & 0 \\ 0 & 0 & 1 \end{pmatrix}$ Define $M = P Q P^T$ and $N = P Q P^\theta$ , where $P^T$ is the transpose of $P$ . If $Z = \frac{\det(M)}{2 \det(N)} N$ , find the value of $\sum_{k=1}^{5} \text{Tr}(Z^k)$

Options

A.
$-3$
B.
$3$
C.
$-2$
D.
$0$

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

Transpose of Product|Euler's Form and Rotations|Square Root and Modulus Logic

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between the transpose ( $P^T$ ) and the conjugate transpose ( $P^\theta$ ) of a complex matrix, specifically within determinant properties.

Students often incorrectly assume $\det(P^T) = \det(P^\theta)$ or blindly apply real matrix properties to complex matrices.

The trap is set by requiring the evaluation of $\det(M) = \det(P)^2$ versus $\det(N) = |\det(P)|^2$ .

A student falling for the trap will find $\det(M) = \det(N)$ , missing the negative sign from $i^2$ , leading to the incorrect sum of $3$ .

Furthermore, it tests the geometric series of traces of rotation matrices in Euler's form.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Transpose of Product
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free