Page 344 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Matrices 335

4. The value of a determinant is unchanged if the rows are written as columns. Thus,
T
the determinants of a matrix A and its transpose matrix A are equal.
5. The value of a determinant is unchanged if, to each element of one row (column) is
added a constant k times the corresponding element of another row (column). Thus,
we have, for example

a 11 a 12 a 11 ‡ ka 12 a 12 a 11 ‡ ka 31 a 12 ‡ ka 32
a 13 a 13 a 13 ‡ ka 33
ˆ
a 21 a 22 ˆ a 21 ‡ ka 22 a 22 a 21 a 22 a 23
a 23 a 23

a 31 a 32 a 33 a 31 ‡ ka 32 a 32 a 33 a 31 a 32 a 33
Each element a ij of the determinant jAj in equation (I.19) has a cofactor C ij , which
is an (n ÿ 1)-order determinant. This cofactor C ij is constructed by deleting the ith
row and the jth column of jAj and then multiplying by (ÿ1) i‡ j . For example, the
cofactor of the element a 12 in equation (I.22) is

a 21 a 23
C 12 ˆÿ ˆ a 23 a 31 ÿ a 21 a 33

a 31 a 33
The summation on the right-hand side of equation (I.20) may be expressed in terms
of the cofactors of the ®rst row of jAj, so that (I.20) becomes
n
X
jAjˆ a 11 C 11 ‡ a 12 C 12 ‡ ‡ a 1n C 1n ˆ a 1k C 1k (I:23)
kˆ1
Alternatively, the expression of jAj in equation (I.20) may be expanded in terms of any
row i
n
X
jAjˆ a ik C ik , i ˆ 1, 2, .. . , n (I:24)
kˆ1
or in terms of any column j
n
X
jAjˆ a kj C kj , j ˆ 1, 2, ... , n (I:25)
kˆ1
Equations (I.20), (I.24), and (I.25) are identical; they are just expressed in different
notations.
Now suppose that row 1 and row i of the determinant jAj are identical. Equation
(I.23) then becomes
n
X
jAjˆ a i1 C 11 ‡ a i2 C 12 ‡ ‡ a in C 1n ˆ a ik C 1k ˆ 0
kˆ1
where the determinant jAj vanishes according to property 3. This argument applies to
any identical pair of rows or any identical pair of columns, so that equations (I.24) and
(I.25) may be generalized
n n
X X
a ik C jk ˆ a ki C kj ˆjAjä ij , i, j ˆ 1, 2, .. . , n (I:26)
kˆ1 kˆ1
2
It can be shown that the determinant of the product of two square matrices of the
same order is equal to the product of the two determinants, i.e., if C ˆ AB, then
:
jCjˆjAj jBj (I:27)
2 See G. D. Arfken and H. J. Weber (1995) Mathematical Methods for Physicists, 4th edition (Academic
Press, San Diego), p. 169.