Consider the matrix M=(1110)M = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}M=(1110). If (anan−1)=Mn(a0a−1)\begin{pmatrix} a_n \\ a_{n-1} \end{pmatrix} = M^n \begin{pmatrix} a_0 \\ a_{-1} \end{pmatrix}(anan−1)=Mn(a0a−1), what is the trace of MnM^nMn?
FnF_nFn
LnL_nLn (Lucas numbers)
Fn−1+Fn+1F_{n-1} + F_{n+1}Fn−1+Fn+1
Fn2F_n^2Fn2