Thus it appears that if two adjacent columns, or rows, of the determinant are interchanged, the sign of the determinant is changed, but its value remains unaltered.
If for the sake of brevity we denote the determinant
by (a_1b_2c_3), then the result we have just obtained may be
written
Similarly we may show that
.
548. Vanishing of a Determinant. If two rows or two columns of the determinant are identical the determinant vanishes.
For let D be the value of the determinant, then by interchanging two rows or two columns we obtain a determinant whose value is -D; but the determinant is unaltered; hence D= - D, that is D=0. Thus we have the follow- ing equations,
549. Multiplication of a Determinant. If each constituent
in any row, or in any column, is multiplied by the same factor,
then the determinant is multiplied by that factor.
For which proves the proposition.
Cor. If each constituent of one row, or column, is the same multiple of the corresponding constituent of another row, or column, the determinant vanishes.
