geometricallythe determinant of square matrix [ an is the (signedvolume of the parallelepiped ( -dimensional parallelogramdefined by the columns an that isthe set of points ni= ai ai where ai the easiest way to compute determinant of general matrix is to apply simple operations to the matrix that potentially reduce its complexity (as in the number of non-zero elementsfor example)while retaining its determinantadding multiple of one column (or rowto anotherdoes not change the determinant multiplying column (or rowwith number multiplies the determinant by the same number swapping two rows changes the sign of the determinant by applying these rules repeatedly one can reduce any matrix to diagonal matrix it follows then that the determinant of the original matrix is equal to the product of the diagonal elements of the resulting diagonal matrix multiplied by known constant example (determinant and volumefigure illustrates how the determinant of matrix can be viewed as signed volumewhich can be computed by repeatedly applying the first rule above herewe wish to compute the area of red parallelogram determined by the matrix given in ( in particularthe corner points of the parallelogram correspond to the vectors [ ][ ][ ]and [ ] - figure the volume of the red parallelogram can be obtained by number of shear operations that do not change the volume adding - times the first column of to the second column gives the matrix -