Comment by RossBencina

Form a quadratic equation to solve for the eigenvalues x of a 2x2 matrix (|A - xI| = 0). The inverse of a matrix can be calculated as the classical adjugate multiplied by the reciprocal of the determinant. Use Cramer's Rule to solve a system of linear equations by computing determinants. Reason that if x is an eigenvalue of A then A - xI has a non-trivial nullspace (using the mnemonic |A - xI| = 0).