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Question #40938

Difficulty: 8/10 | 🟢 Free

The Question

Let $B_n = \begin{pmatrix} \sum_{r=1}^n \frac{1}{\sqrt{n^2+r}} & \sum_{r=1}^n \frac{r^2}{n^3+r^2} \\ \sum_{r=1}^n \frac{r}{n^2+r} & \sum_{r=1}^n \frac{1}{\sqrt{4n^2+r}} \end{pmatrix}$ and $A = \lim_{n \to \infty} (3I - B_n)^{-1}$ , where $I$ is the $2 \times 2$ identity matrix. Find the value of $\lim_{k \to \infty} \det(A^k)$ .

Options

A.
$0$
B.
$1$
C.
$\frac{6}{29}$
D.
$\infty$

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

Squeeze Theorem (Sandwich Theorem)|Inversion and Determinant Product

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Squeeze Theorem on matrix elements and the properties of determinants.

The primary trap is the 'inverse matrix distraction': students often waste time explicitly calculating the inverse of $(3I - B)$ and then attempting to find a pattern for $A^k$ .

The optimal path recognizes that $\det(A^k) = (\det A)^k$ .

By bounding the elements using the Squeeze Theorem, we find $\det(3I - B) = 29/6$ , meaning $\det(A) = 6/29 < 1$ .

Consequently, the limit as $k \to \infty$ is trivially 0.

It also reinforces that if a matrix has eigenvalues strictly between -1 and 1, its powers converge to the zero matrix.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Squeeze Theorem (Sandwich Theorem)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free