Matrix Theory - Isometries

Find the eigenvalues and eigenvectors of A.
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Consider the matrices in O(2) of the form:

;

these matrices correspond to the elements of O(2) with det(A)= -1. Find the eigenvalues and eigenvectors of A.

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...1:
we need to find all (u,v) s.t.:
A(u,v)=x1(u,v) which means:
cost*u+sint*v=u
sint*u-cost*v=v
which can be written as:
(cost-1)*u+sint*v=0
sint*u-(cost+1)*v=0
From the first equation we get: v=-(cost-1)/sint*u,
and because the determinant of the system is -(cost-1)(cost+1)-sint*sint=0 we know that the second equation is just a scalar multiple of the first, so ...