IMC 2006 · Problem 6

Let $A_i$, $B_i$, $S_i$ ($i = 1, 2, 3$) be invertible real $2 \times 2$ matrices such that

(1) not all $A_i$ have a common real eigenvector;

(2) $A_i = S_i^{-1} B_i S_i$ for all $i = 1, 2, 3$;

(3) $A_1 A_2 A_3 = B_1 B_2 B_3 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$.

Prove that there is an invertible real $2 \times 2$ matrix $S$ such that $A_i = S^{-1} B_i S$ for all $i = 1, 2, 3$.