Question

. The Fibonacci sequence is a sequence given by the equations: F1 = 1, F2 = 1, Fn+2 = Fn+1 + Fn So for example, F3 = F2 + F1 =1+1=2 F1 = F3+ F2 = 2 +1 = 3 (a) (2 marks) Use the definition of the (n +1)t Fibonacci number to find a 2 × 2matrix A which satisfies the equation: A\left(\begin{array}{c} F_{n} \\ F_{n+1} \end{array}\right)=\left(\begin{array}{c} F_{n+1} \\ F_{n+2} \end{array}\right) (b) (8 marks) Compute the eigenvalues of the matrix A that you found in the previous question. For each eigenvalue, find a basis for the eigen space. (c) (2 marks) Find an invertible matrix P and aA = PDP-1diagonal matrix D such that (d) 4 marks Use the equation from part (a) to prove by induction on n that for alln > 0, A^{n}\left(\begin{array}{l} F_{1} \\ F_{2} \end{array}\right)=\left(\begin{array}{l} F_{n+1} \\ F_{n+2} \end{array}\right) (e) (4 marks) Use parts (c) and (d) to prove following formula for the nth Fibonacci number: F_{n}=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right)  Fig: 1  Fig: 2  Fig: 3  Fig: 4  Fig: 5  Fig: 6  Fig: 7  Fig: 8  Fig: 9  Fig: 10  Fig: 11  Fig: 12  Fig: 13