Example 2

Let $\sigma$ be a permutation of the interval $[1, n]$ in $\mathbf{N}$, with signature $\varepsilon (\sigma )$. Let M$(\sigma )$ be the matrix of the permutation $\sigma$ (II, §10, No. 7, p. 351). The columns of the matrix M$(\sigma )$ are $\varepsilon_{\sigma(1)}, . . . , \varepsilon_{\sigma(n)}$. Applying formula (4) of VIII, p. 448, we therefore have

(22) det(M$(\sigma )$) $=\pi (\varepsilon (\sigma ))$.