Lecture Notes

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 $\sigma(i) > \sigma(i+1)$.