linear algebra-permutation

Question

Given the permutation

$$\sigma = \begin{pmatrix} 1&2&3&4&5\\3&1&2&5&4\end{pmatrix}$$

the matrix A is defined to be the one whose i-th column is the $\sigma(i)$-th column of the identity I. Which of the following is correct?

1. $A=A^{-2}$

2. $A=A^{-4}$

3. $A=A^{-5}$

4. $A=A^{-1}$

I didn't understand what $\sigma(i)$-th means here?

Answer 1

We can think of a permutation on $S := \{1, 2, 3, 4, 5\}$ as a bijective map $S \to S$, in which case $\sigma(i)$ is the usual function notation. For your $\sigma$, $\sigma(1) = 3$, $\sigma(2) = 1$, etc.

Answer 2

$$\sigma{(1)}=3, \sigma{(2)}=1, \sigma{(3)}=2, \sigma{(4)}=5, \sigma{(5)}=4$$

Answer 3

$$\sigma = \begin{pmatrix} 1&2&3&4&5\\3&1&2&5&4\end{pmatrix}$$

$$\sigma(1) = 3, \;\\ \sigma(2) = 1,\;\\ \sigma(3) = 2,\;\\ \sigma(4) = 5,\;\\ \sigma(5) = 4.$$

$$A = \begin{pmatrix} 0&1&0&0&0\\0&0&1&0&0\\ 1&0&0&0&0\\ 0&0&0&0&1\\ 0&0&0&1&0\end{pmatrix}$$