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Prove that two 2*2 matrices have common eigenvector

Source: 2016 South Korea USCM P6

August 16, 2020

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Problem Statement

A

and

B

are

2\times 2

real valued matrices satisfying \det A = \det B = 1, \text{tr}(A)>2, \text{tr}(B)>2, \text{tr}(ABA^{-1}B^{-1}) = 2 Prove that

A

and

B

have a common eigenvector.