** quiz-style questions:** 
    * Find the sign of the permutation $(4312)$.
    * Compute $\det(A - \lambda I)$ where $$A = \begin{bmatrix} 1 & 0 & -1\cr 0 & 1 & -1\cr -1 & 1 & 0 \end{bmatrix}$$ 
    * Let $\sigma$ be a permutation of length $\ell > 0$. Show that there must exist some $i$ such that $\sigma(i) > \sigma(i+1)$.