so what i'm trying to show is that the set of all 3x3 matrices with determinant = 1, such that for a fixed vector r, Ar = r is isomorphic to the set of all 2x2 matrices with determinant = 1.
since r is an eigenvector with eigenvalue 1 of the 3x3 matrix, the other 2 eigenvalues of the 3x3 matrix must multiply to 1 and i was thinking about how to map each 3x3 matrix with eigenvalues 1, , to a 2x2 matrix with eigenvalues and . however i am having trouble coming up with the explicit mapping. I tried taking a 3x3 matrix with those specified eigenvalues and thought about how to "trim the edges" (for example, remove the first column and first row) to get a 2x2 matrix with the same eigenvalues as the eigenvalues besides the 1 of the 3x3 matrix. but this process may "mess up" the determinant for some 3x3 matrices and end up creating a 2x2 matrix with different eigenvalues then the one i wanted. furthermore i don't even think that this is well defined.
can someone give me some hints in the right direction on how to define such a function from one set to the other? thanks.