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Question #59682

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a $3 \times 3$ real matrix with three distinct real eigenvalues in an arithmetic progression. Let $S$ be a dataset consisting of the values of the function $f(x) = \text{tr}(\text{adj}(xI - A))$ evaluated at each of the three eigenvalues of $A$ . If $MD_{\text{mean}}$ and $MD_{\text{median}}$ denote the mean deviation about the mean and the mean deviation about the median of the dataset $S$ respectively, which of the following expressions identically equals $MD_{\text{mean}} - MD_{\text{median}}$ ?

Options

A.
$\frac{1}{6} \left( \text{tr}(A^2) - \frac{1}{3}(\text{tr}(A))^2 \right)$
B.
$\frac{1}{3} \left( \text{tr}(A^2) - \frac{1}{3}(\text{tr}(A))^2 \right)$
C.
$\frac{1}{2} \left( \text{tr}(A^2) - \frac{1}{3}(\text{tr}(A))^2 \right)$
D.
$\frac{2}{3} \left( \text{tr}(A^2) - \frac{1}{3}(\text{tr}(A))^2 \right)$

Check Your Answer

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

Mean Deviation about Mean vs. Median|Differentiation of Determinants|Characteristic Equation

Why This Question Is Tricky

Examiner Levers

Negative Thinking

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

Difficulty Boost : +2

Logical Assertion

Tests the causal relationship between two true statements.

Difficulty Boost : +4

Boundary Condition

Tests behavior at limits like 0, 1, infinity, or empty sets.

Difficulty Boost : +4

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This problem tests the intersection of spectral theory, differential properties of determinants, and statistical measures.

The primary trap lies in the evaluation of $f(x) = \text{tr}(\text{adj}(xI - A))$ .

Students unaware of Jacobi's formula ( $d/dx \det(M) = \text{tr}(\text{adj}(M) \cdot M')$ ) will attempt a brute-force expansion of the adjugate matrix, leading to an O(n!) algebraic nightmare.

By recognizing $f(x)$ as the derivative of the characteristic polynomial, the dataset is rapidly generated.

Furthermore, the problem enforces pure symbolic manipulation by requiring the final statistical difference to be expressed in terms of matrix invariants ( $\text{tr}(A)$ and $\text{tr}(A^2)$ ), testing the student's ability to link roots of polynomials back to trace properties without relying on numerical values.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Mean Deviation about Mean vs. Median
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free