1 (5 x 10-6 x 8) -2 (4 x 10-6 x7) +3 (4 x 8-5 x 7) =
The amount of calculations required to expand a determinant increases very quickly with increasing order n. This is because the minor of each cofactor must be expanded, the latter producing new cofactors whose minors are in turn expanded, and so forth until cofactors of order 2 are reached. Another, faster method is normally used to calculate determinants by computer. Before describing this method, however, some properties of determinants must be examined; in all cases, column may be substituted for row.
(1) The determinant of a matrix is equal to that of its transpose, since a determinant may be computed from either the rows or columns of the matrix: | A' | = | A |.
(2) If two rows are interchanged, the sign of the determinant is reversed.
(3) If two rows are identical, the determinant is null (corollary of the second property; see beginning of the present Section).
(4) If a scalar is a factor of one row, it becomes a factor of the determinant (since it appears once in each product).
(5) If a row is a multiple of another row, the determinant is null (corollary of properties 4 and 3, i.e. factoring out the multiplier produces two identical rows).
(6) If all elements of a row are 0, the determinant is null (corollary of property 4).
(7) If a scalar c is a factor of all rows, it becomes a factor cn of the determinant (corollary of property 4), i.e. | cB | = cn | B |.
(8) If a multiple of a row is added to another row, the value of the determinant remains unchanged.
(9) The determinant of a triangular matrix (and therefore also of a diagonal matrix) is the product of its diagonal elements.
(10) The sum of the products of the elements of a row with the corresponding cofactors of a different row is equal to zero.
Properties 8 and 9 can be used for rapid computer calculation of the value of a Pivotal determinant; the method is called pivotal condensation. The matrix is first reduced to condensation triangular form using property 8. This property allows the stepwise elimination of all terms on one side of the diagonal through combinations of multiplications by a scalar, and addition and subtraction of rows or columns. Pivotal condensation may be performed in either the upper or the lower triangular parts of a square matrix. If the lower triangular part is chosen, the upper left-hand diagonal element is used as the first pivot to modify the other rows in such a way that their left-hand terms become zero. The technique consists in calculating by how much the pivot must be multiplied to cancel out the terms in the rows below it; when this value is found, property 8 is used with this value as multiplier. When all terms under the diagonal element in the first column are zero, the procedure is repeated with the other diagonal terms as pivots, to cancel out the elements located under them in the same column. Working on the pivots from left to right insures that when values have been changed to 0, they remain so. When the whole lower triangular portion of the matrix is zero, property 9 is used to compute the determinant which is then the product of the modified diagonal elements.
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