$$
\Psi(x_1, x_2, \ldots, x_A)
= \frac{1}{\sqrt{N!}}
\det \left |
\begin{array}{cccc}
\phi_1(x_1) & \phi_1(x_2) & \ldots & \phi_1(x_A) \\
\phi_2(x_1) & \phi_2(x_2) & \ldots & \phi_2(x_A) \\
\vdots & & & \\
\phi_A(x_1) & \phi_A(x_2) & \ldots & \phi_A(x_A)
\end{array}
\right |
$$
Properties of the determinant (interchange of any two rows or
any two columns yields a change in sign; thus no two rows and no
two columns can be the same) lead to the Pauli principle:
- No two particles can be at the same place (two columns the same); and
- No two particles can be in the same state (two rows the same).