The determinant is a value that can be computed from the elements of a square matrix. It is useful in the analysis of systems of linear equations. For a $3\times 3$ matrix, the determinant is defined as \begin{equation*} (x_1, x_2, x_3; y_1, y_2, y_3; z_1, z_2, z_3) \\ =x_1(y_2z_3-y_3z_2)-x_2(y_1z_3-y_3z_1)+x_3(y_1z_2-y_2z_1) \end{equation*}

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