Before You Begin

Question #59512

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{x, y, z\}$ be a data set of real numbers. A matrix $A = \begin{bmatrix} x & y & z \\ y & z & x \\ z & x & y \end{bmatrix}$ is constructed such that $A$ is an orthogonal matrix with $\det(A) > 0$ . Let $\sigma^2$ be the variance of $S$ . A student ignores the orthogonality condition and calculates the maximum possible variance of $S$ , denoted as $V_0$ , under the assumption that $A$ is a singular matrix satisfying $\text{Tr}(A A^T) = 3$ . What is the value of $9(\sigma^2 + V_0)$ ?

Options

A.
3
B.
4
C.
5
D.
6

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

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

Why This Question Is Tricky

Examiner Levers

Negative Thinking

Uses "NOT", "EXCEPT", or "INCORRECT" to flip the logic gate.

Difficulty Boost : +2

Extraneous Data

Provides unnecessary variables (Red Herrings) to test conceptual focus.

Difficulty Boost : +3

Concept Bridge

Cross-chapter synthesis (e.g., using Logarithms in Electrochemistry).

Difficulty Boost : +5

Common Mistakes

This question tests the rigid constraints of orthogonal matrices against statistical variance.

The trap lies in the student's hypothetical scenario: forcing a matrix structure to be singular (det(A)=0) while maintaining the trace condition.

While mathematically impossible for a true orthogonal matrix (where det(A) = ±1), the problem forces the candidate to evaluate the domain constraints under this false assumption to find V_0.

The extraneous condition det(A) > 0 acts as a red herring for the true variance, as the variance depends only on the squared sum of the elements, which is invariant to the sign of the determinant.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Standard Deviation and Coefficient of Variation
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free