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Question #40710

Difficulty: 8/10 | 🟢 Free

The Question

Let $\alpha_n$ and $\beta_n$ be the roots of the quadratic equation $x^2 - n x + 1 = 0$ . For each integer $n \ge 0$ , define the matrix $A_n = \begin{pmatrix} \alpha_n + \beta_n & \alpha_n & \beta_n \\ \alpha_n & \alpha_n + \beta_n & \beta_n \\ \beta_n & \alpha_n & \alpha_n + \beta_n \end{pmatrix}$ Find the value of $\sum_{n=0}^{10} \det(\text{adj}(A_n) + I)$ where $I$ is the $3 \times 3$ identity matrix.

Options

A.
4642
B.
4640
C.
4641
D.
4620

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

Quadratic Roots in Sequences|Adjoint-Determinant Relationship|Singular vs Non-Singular Matrices

Why This Question Is Tricky

Common Mistakes

This question tests the spectral properties of the adjoint matrix, specifically targeting the trap where students incorrectly assume $\text{adj}(A) = |A|A^{-1}$ .

For $n=0$ , the matrix $A_0$ is singular ( $|A_0|=0$ ), making $A_0^{-1}$ undefined.

Students relying on the inverse formula will either skip the $n=0$ term (yielding 4640) or incorrectly assume $\text{adj}(A_0)$ is the zero matrix (yielding 4641).

The rigorous approach requires recognizing that the eigenvalues of $\text{adj}(A)$ are the products of pairs of eigenvalues of $A$ , a property that holds continuously even when the matrix is singular.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Quadratic Roots in Sequences
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free