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\section{Generating Operations}  In some sense, the determinant is able to generate the features if of  various algebras we know and love, For example vectors and their relationships, we have \begin{equation} \textbf{r}_1 \times \textbf{r}_2 = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \\ x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \end{vmatrix} \\ \\