A simple \( 2\times 2 \) determinant illustrates this. We have $$ det(\mathbf{A})= \left| \begin{array}{cc} a_{11}& a_{12}\\ a_{21}&a_{22}\end{array} \right|= (-1)^0a_{11}a_{22}+(-1)^1a_{12}a_{21}, $$ where in the last term we have interchanged the column indices \( 1 \) and \( 2 \). The natural ordering we have chosen is \( a_{11}a_{22} \).