Lecture Notes

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quiz-style questions:

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