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Question #40481

Difficulty: 8/10 | 🟢 Free

The Question

Let $B = \begin{pmatrix} 1 & -\sqrt{3} & 2+\sqrt{3} \\ \sqrt{3} & 1 & 2-\sqrt{3} \\ 0 & 0 & 3 \end{pmatrix}$ . For each $n \in \mathbb{N}$ , define the matrix $M_n = B^n - 2^n I$ , where $I$ is the $3 \times 3$ identity matrix. Let $S$ be the set of all integers $n \in \{1, 2, \dots, 100\}$ such that $\det(\text{adj}(M_n)) > 0$ . Find the sum of all elements in the set $S$ .

Options

A.
4234
B.
5050
C.
4134
D.
4324

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

De Moivre’s Theorem Applications|Characteristic Equation|Singular vs Non-Singular Matrices

Why This Question Is Tricky

Common Mistakes

This question tests the application of De Moivre's Theorem to find the eigenvalues of a matrix power, combined with the properties of the adjoint matrix.

The specific trap lies in the assumption that $B^n - 2^n I$ is always non-singular because $B$ has no eigenvalue equal to 2 (its eigenvalues are 3 and $1 \pm i\sqrt{3}$ ).

A student might falsely assume that if $B - 2I$ is non-singular, then $B^n - 2^n I$ is also non-singular for all $n$ , leading them to include all integers from 1 to 100 (Option B).

However, because the complex eigenvalues of $B$ have a magnitude of exactly 2, their $n$ -th powers align with $2^n$ at specific intervals ( $n=6k$ ), causing the matrix $M_n$ to become singular and collapsing the determinant to zero.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	De Moivre’s Theorem Applications
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free