Mathematical Methods for Physicists: A concise introduction - Site Map

APPENDIX 2 DETERMINANTS
D ˆ a i1 C i1 ‡ a i2 C i2 ‡ ‡a in C in
ˆ Xn
kˆ1
a ik C ik …i ˆ 1; 2; ...; or n† …A2:10a†
…cofactor expansion along the ith row†
or
D ˆ a 1k C 1k ‡ a 2k C 2k ‡‡a nk C nk
ˆ Xn
iˆ1
a ik C ik …k ˆ 1; 2; ...; or n†: …A2:10b†
…cofactor expansion along the kth column†
We see that D is de®ned in terms of n determinants of order n 1, each of
which, in turn, is de®ned in terms of n 1 determinants of order n 2, and so on;
we ®nally arrive at second-order determinants, in which the cofactors of the
elements are single elements of D. The method of evaluating a determinant just
described is one form of Laplace's development of a determinant.
Problem A2.3
For a second-order determinant
D ˆ a11 a 12
a 21 a 22
show that the Laplace's development yields the same value of D no matter which
row or column we choose.
Problem A2.4
Let
1 3 0
D ˆ
2 6 4
:
1 0 2
Evaluate D, ®rst by the ®rst-row expansion, then by the ®rst-column expansion.
Do you get the same value of D?
Properties of determinants
In this section we develop some of the fundamental properties of the determinant
function. In most cases, the proofs are brief.
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