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Question #40978

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a $5 \times 5$ real matrix of rank $3$ such that $\text{Tr}(A) = 20$ and $A^4 - 20A^3 + 131A^2 - 280A = O$ Let $S$ be the data set consisting of all the eigenvalues of $A$ (including multiplicities). If $M_1$ and $M_2$ are the mean deviations of $S$ about its mean and median respectively, then the value of $45(M_1 - M_2)$ is:

Options

A.
9
B.
5
C.
12
D.
18

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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|Matrix Trace Properties|Linear Dependency of Rows

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of linear algebra and statistics, specifically targeting the trap of ignoring zero eigenvalues in a rank-deficient matrix.

Students often find the non-zero roots of the annihilating polynomial (5, 7, 8) and incorrectly assume these form the entire data set, leading to an erroneous mean and median (yielding the trap answer of 5).

By explicitly stating the matrix is 5x5 with rank 3, the student must deduce that the eigenvalue 0 has an algebraic multiplicity of 2.

Including these zeros fundamentally shifts both the mean and the median, testing whether the student understands that 'all eigenvalues' of an n x n matrix must total n elements, and that zero is a valid data point in statistical dispersion measures.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Mean Deviation about Mean vs. Median
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free