Page 546 - Modelling in Transport Phenomena A Conceptual Approach

526 APPENDIX A. MATHEMATICAL PIUZJMINARIES

A.9.2.1 Some properties of determinants

1. If all elements in a row or column are zero, the determinant is zero, i.e.,
a1 bl c1
(A.418)
0 0 0

2. The value of a determinant is not altered when the rows are changed to
columns and the columns to rows, i.e., when the rows and columns are inter-
changed.

3. The interchange of any two columns or any two rows of a determinant changes
the sign of the determinant.

4. If two columns or two rows of a determinant are identical, the determinant
is equal to zero.

a1 +dl bl c1 a1 bl c1 dl bl c1
a2 b2 + 4 bz c2 (A.9-19)
a3+d3 b3 c3 a3 4 c3 d3 4 c3

6. Adding the same multiple of each element of one row to the corresponding
element of another row does not change the value of the determinant. The
same holds true for the columns.

(A.9-20)

This result follows immediately from Properties 4 and 5.

7. If all the elements in any column or row are multiplied by any factor, the
determinant is multiplied by that factor, i.e.,

(A.9-21)

(A.9-22)