Consider the permutation \(\sigma\) defined by \(1 \mapsto 3,\) \(2\mapsto 1,\) \(3\mapsto 4\) and \(4\mapsto 2.\)

Show that the transpose of a permutation matrix is its inverse.

Compute the adjugate matrix of the following matrices \[\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix},\quad \begin{pmatrix} 1 & 1 & 2 \\ 2 & 4 & 2 \\ 0 & 2 & 1 \end{pmatrix}, \quad \begin{pmatrix} 4 & -1 & 1 \\ 1 & 1 & -2 \\ 1 & -1 & 1 \end{pmatrix}.\]

Let \(n \in \mathbb{N}\) and \(\mathbf{A}=(A_{ij})_{1\leqslant i,j\leqslant n} \in M_{n,n}(\mathbb{R})\) be a square matrix with \(A_{ij}\) all integers. Show that the entries of \(\mathbf{A}^{-1}\) are all integers if and only if \(\det(\mathbf{A})=\pm 1.\)