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Question 2.7.13: (a) Find a 3 by 3 permutation matrix with P^3 = I (but not P...

(a) Find a 3 by 3 permutation matrix with {P}^{3} = I (but not P = 1).
(b) Find a 4 by 4 permutation \hat{P} with \hat{P}^{4} \neq I.

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(a) A cyclic P=\left[ \begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix} \right] or its transpose will have {p}^{3} = I:

(1, 2, 3)\rightarrow (2, 3, 1)\rightarrow (3, 1, 2) \rightarrow (1, 2, 3).

 (b) \hat{P}=\begin{bmatrix} 1 & 0 \\ 0 & P \end{bmatrix} for the same P has 

\hat{P}^{4}=\hat{P} \neq I.

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