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Linear Algebra : Invertible Matrices

Let A be the real 2x2 matrix
[a b]
[c d]

with bc greater than or equal to 0. Prove there exists a real 2x2 invertible matrix S so that S^-1 A S is either diagonal or of the form
[x 1]
[0 x]
where x is the eigenvalue of A.

Solution Summary

The link between invertibility of matrices and eigenvalues is investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who posted the question.