$$\displaystyle \det A = \sum_{\sigma \in \mathcal S_n}\varepsilon(\sigma)a_{\sigma(1),1}a_{\sigma(2),2}\cdots a_{\sigma(n),n} $$
$$\displaystyle \det A = \sum_{\sigma \in \mathcal S_n}\varepsilon(\sigma)a_{\sigma(1),1}a_{\sigma(2),2}\cdots a_{\sigma(n),n} $$
$$\displaystyle \quad n=\sum _{d|n}\varphi (d)$$