For a 2 2 matrix A , whose elements are real numbers, denote by A^m the product AA A _ m times , where m is a positive integer. Define x₀ = 0 , x₁ = 1 , x_n = x_ n-1 + x_ n-2 , for all n 2 and A_n = bmatrix x_ n+1 & x_n x_n & x_ n-1 bmatrix , for all n 1 . Which of the following statements is TRUE for all m 3 ?

Step-by-step solution

Given x₀ = 0 , x₁ = 1 , and x_n = x_ n-1 + x_ n-2 for n 2 . For n = 1 , the matrix A₁ is: A₁ = bmatrix x₂ & x₁ x₁ & x₀ bmatrix = bmatrix 1 & 1 1 & 0 bmatrix The characteristic equation of A₁ is given by |A₁ - I| = 0 : vmatrix 1- & 1 1 & - vmatrix = 0 - (1- ) - 1 = 0 ^2 - - 1 = 0 By the Cayley-Hamilton theorem, every square matrix satisfies its own characteristic equation. Therefore: A₁^2 - A₁ - I = 0 A₁^2 = A₁ + I Multiplying both sides by A₁^ m-2 (for m 3 ), we get: A₁^m = A₁^ m-1 + A₁^ m-2 Checking the other opti