I have two known types of matrices, $$L_j= \left[ \begin{matrix} 1 & 1 \\ f_j & -f_j \end{matrix} \right]$$ and $$P_j= \left[ \begin{matrix} e^{i a_j k_j} & 0 \\ 0 & e^{-i a_j k_j} \end{matrix} \right]$$
I have another matrix defined by $T_j=L_j \cdot P_j \cdot L_j^{-1}$
I have to find the trace of a total matrix defined by $$M_n= T_n \cdot T_{n-1} \dot \,\, ...\,\cdot T_1 $$ for a known $n$.
So far, I know that $ Tr (T_j ) = Tr (L_j) =2\cos (a_j k_j)$ because they are similar. I also know that any cyclic permutation of the matrices in a product of matrices does not effect the trace of the product. I also noted that $$ Tr(P_1\cdot P_2\cdot \,\,...\,\cdot P_n)=e^{i \sum_{j=1}^n a_j k_j }+e^{-i \sum_{j=1}^n a_j k_j } =2\cos (\sum_{j=1}^n a_j k_j) $$ I'm stuck here. I calculated the traces for $M_2$ and $M_3$ and it does intuitively seem as though the terms are somewhat related, although I cannot say how. Are there any useful properties of matrices I'm missing here? Or any suggestions on how to approach this problem?