Proving Fibonacci Identity

Question

in this question I am asked to prove that if $A=\begin{bmatrix}1 & 1 \\ 1 & 0 \end{bmatrix}$ then $A^n= \begin{bmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{bmatrix}$ which I have successfully proved by induction on n. Then we are asked to prove that $F_{k+j} = F_kF_{j+1} + F_{k−1}F_j = F_{k+1}F_j + F_kF_{j−1}$ by comparing the identity $A^{k+j}= A^kA^j $ And then a similar question for $A^{k+j+l}= A^kA^jA^l $ , I have tried answering these questions by computing the matrix identities but it got too complicated and I got lost in the algebra, is there an easier way to prove this? Thank you!

Answer 1

$$\begin{align}A^{k+j}&= \begin{bmatrix} F_{k+j+1} & \color{red}{F_{k+j}} \\ \color{blue}{F_{k+j}} & F_{k+j-1} \end{bmatrix}\\A^kA^j= \begin{bmatrix} F_{k+1} & F_k \\ F_k & F_{k-1} \end{bmatrix}\begin{bmatrix} F_{j+1} & F_j \\ F_j & F_{j-1} \end{bmatrix}&=\begin{bmatrix} F_{k+1}F_{j+1}+F_kF_j & \color{red}{F_{k+1}F_j+F_kF_{j-1}} \\ \color{blue}{F_kF_{j+1}+F_{k-1}F_j} & F_kF_j+F_{k-1}F_{j-1} \end{bmatrix}\end{align}$$