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This follows from the identities $$A_1^{-1}=UA_1U^{-1},\;\;A_2^{-1}=UA_2U^{-1},$$ with $$U=U^{-1}=\left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{array} \right).$$ Hence if $\lambda\neq 0$ is an eigenvalue of any string of products of the two matrices $A_1$ and $A_2$, then also $1/\lambda$ is an eigenvalue.