We start by finding the eigenvalue. We know this equation must be true: Next we put in an identity matrix so we are dealing with matrix-vs-matrix: Bring all to left hand side: If v is non-zero then we can (hopefully) solve for λ using just the determinant: Let's try that equation on our previous example: Start with | A − λI | = 0. Which is:
Solve the characteristic polynomial for the eigenvalues. This is, in general, a difficult step for finding eigenvalues, as there exists no general solution for quintic functions or higher polynomials. However, we are dealing with a matrix of dimension 2, so the quadratic is easily solved.