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Question #40994

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix}$ be a real orthogonal matrix. Let $S$ be the data set consisting of the 9 entries of $A$ . If $\sigma^2$ and $\mu$ are the variance and mean of $S$ respectively, and $V = \frac{\sigma}{|\mu|}$ , which of the following statements is true?

Options

A.
$V = \sqrt{2}$ and $\det(A) \in \{-1, 1\}$
B.
$V = \sqrt{3}$ and $\det(A) \in \{-1, 1\}$
C.
$V$ is undefined because $\mu = 0$ and $\det(A) = 0$
D.
$V = \frac{1}{\sqrt{2}}$ and $\det(A) \in \{-1, 1\}$

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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

Standard Deviation and Coefficient of Variation|Orthogonal Matrix Conditions|Singular vs Non-Singular Matrices

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of matrix algebra and statistics.

The core trap lies in the assumption that a symmetric-looking matrix with variables can be singular ( $\det(A)=0$ ), which would lead to $x+y+z=0$ and an undefined coefficient of variation.

However, the rigid geometric constraint of an orthogonal matrix ( $AA^T=I$ ) forces the determinant to be $\pm 1$ and strictly bounds the sum of the elements.

Students who fail to apply the orthonormal column conditions ( $x^2+y^2+z^2=1$ and $xy+yz+zx=0$ ) will incorrectly deduce that $\mu=0$ is possible, falling for the domain error trap.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Standard Deviation and Coefficient of Variation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free