Let A be a 2x2 matrix with real entries of determinant 1 (so that the correspond

Question

Let A be a 2x2 matrix with real entries of determinant 1 (so that the corresponding linear map preserves the area on the plane. Describe all possible pairs of eigenvalues of A. Conclude that if A is hyperbolic, that is, it has no eigenvalues on the unit circle, then its eigenvalues are both real and one is the inverse of the other. What does this say about the possible types of fixed points of A?

Explanation / Answer

let p be an eigen value hence det(A-pI)=0 let elemtns of A are a11 a12 a21 a22 hence (a11-p)(a22-p)-a12a21=0 => a11a22+p^2 - (a11+a22)p=a12a21 given det A =1 hence a11a22-a12a21=1 therefore,p^2-(a11+a22)+1=0 hence roots of this equation are two eigen values of A...

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