Assess students' ability to compute eigenvalues and eigenvectors of 2x2 real matrices, determine diagonalizability, use diagonalization or eigen-decomposition to compute matrix powers, and apply these techniques to derive closed-form solutions of linear recurrence relations (2-term systems and second-order scalar recurrences) from given initial conditions; include problems with distinct eigenvalues and one scenario addressing repeated or non-diagonalizable cases requiring interpretation and verification.