Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Digital systems, linear algebra, introduction to formal proof / 4 / c / Solution

The initial vector records both initial Fibonacci numbers: $V_1=(U_2,U_1{)}^{\top }=(1,1{)}^{\top }$. For $n\ge 2$,Conversely this vector recurrence forces the second component to be the preceding first component and forces their sum to be the next first component, recovering the scalar recurrence and its initial conditions. Iterating, or using mathematical induction, yields $V_n=A^{n-1}(1,1{)}^{\top }$ for every $n\ge 1$.