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Limit of matrix entries

Source: Mexican University Math Olympiad 2024, Problem 4

October 1, 2024
Matriceslimitlinear algebraseries

Problem Statement

Given b>0 b > 0 , consider the following matrix: B=(bb2b2b3) B = \begin{pmatrix} b & b^2 \\ b^2 & b^3 \end{pmatrix} Denote by ei e_i the top left entry of Bi B^i . Prove that the following limit exists and calculate its value: lim⁡i→∞ei. \lim_{i \to \infty} \sqrt{e_i}.
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