MathDB

The scores of this problem were: one time 17/20 (by the runner-up) one time 4/20 (by Andrei Negut) one time 1/20 (by the winner) the rest had zero... just to give an idea of the difficulty. Let

A_{i},B_{i},S_{i}

(

i=1,2,3

) be invertible real

2\times 2

matrices such that [*]not all

A_{i}

have a common real eigenvector, [*]

A_{i}=S_{i}^{-1}B_{i}S_{i}

for

i=1,2,3

, [*]

A_{1}A_{2}A_{3}=B_{1}B_{2}B_{3}=I

. Prove that there is an invertible

2\times 2

matrix

S

such that

A_{i}=S^{-1}B_{i}S

for all

i=1,2,3

Problem Statement