Mathematical Methods for Physicists: A concise introduction - Site Map

APPENDIX 2 DETERMINANTS
Example A2.2
1 3 0
1 3 0
1 1 0
1 1 0
2 6 4
ˆ 2
1 3 2
ˆ 2 3
1 1 2
ˆ 2 3 2
1 1 1
ˆ12:
1 0 2
1 0 2
1 0 2
1 0 1
(3) The value of a determinant is not altered if its rows are written as columns,
in the same order.
Proof: Since the same value is obtained whether we expand a determinant by
any row or any column, thus we have property (3). The following example will
illustrate this property.
Example A2.3
1 0 2
D ˆ
1 1 0
ˆ 1 1 0
1 3 0 1 0
2 3 ‡ 2 1 1
2 1 ˆ 1:
2 1 3
Now interchanging the rows and the columns, then evaluating the value of the
resulting determinant, we ®nd
1 1 2
0 1 1
ˆ 1 1 1
0 3 …1† 0 1
2 3 ‡ 2 0 1
2 0 ˆ 1;
2 0 3
illustrating property (3).
(4) If any two rows (or two columns) of a determinant are interchanged, the
resulting determinant is the negative of the original determinant.
Proof: The proof is by induction. It is easy to see that it holds for 2 2 determinants.
Assuming the result holds for n n determinants, we shall show that it
also holds for …n ‡ 1†…n ‡ 1† determinants, thereby proving by induction that it
holds in general.
Let B be an …n ‡ 1†…n ‡ 1† determinant obtained from D by interchanging
two rows. Expanding B in terms of a row that is not one of those interchanged,
such as the kth row, we have
B ˆ Xn
jˆ1
…1† j‡k b kj M 0 kj;
where M 0 kj is the minor of b kj . Each b kj is identical to the corresponding a kj (the
elements of D). Each M 0 kj is obtained from the corresponding M kj (of a kj )by
544