Mathematical Methods for Physicists: A concise introduction - Site Map

PROPERTIES OF DETERMINANTS
interchanging two rows. Thus b kj ˆ a kj , and Mkj 0 ˆM kj . Hence
B ˆ Xn
…1† j‡k b kj M kj ˆD:
jˆ1
The proof for columns is similar.
Example A2.4
Consider
1 0 2
D ˆ
1 1 0
ˆ 1:
2 1 3
Now interchanging the ®rst two rows, we have
1 1 0
B ˆ
1 0 2
ˆ1
2 1 3
illustrating property (4).
(5) If corresponding elements of two rows (or two columns) of a determinant
are proportional, the value of the determinant is zero.
Proof: Let the elements of the ith and jth rows of D be proportional, say,
a ik ˆ ca jk ; k ˆ 1; 2; ...; n. Ifc ˆ 0, then D ˆ 0. For c 6ˆ 0, then by property (2),
D ˆ cB, where the ith and jth rows of B are identical. Interchanging these two
rows, B goes over to B (by property (4)). But the rows are identical, the new
determinant is still B. Thus B ˆB; B ˆ 0, and D ˆ 0.
Example A2.5
1 1 2
B ˆ
1 1 0
ˆ 0;
2 2 8
3 6 4
D ˆ
1 1 3
ˆ 0:
6 12 8
In B the ®rst and second columns are identical, and in D the ®rst and the third
rows are proportional.
(6) If each element of a row of a determinant is a binomial, then the determinant
can be written as the sum of two determinants, for example,
4x ‡ 2 3 2
4x 3 2
2 3 2
x 4 3
ˆ
x 4 3
‡
0 4 3
:
3x 1 2 1
3x 2 1
1 2 1
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