Mathematical Methods for Physicists: A concise introduction - Site Map

Appendix 2
Determinants
The determinant is a tool used in many branches of mathematics, science, and
engineering. The reader is assumed to be familiar with this subject. However, for
those who are in need of review, we prepared this appendix, in which the determinant
is de®ned and its properties developed. In Chapters 1 and 3, the reader will
see the determinant's use in proving certain properties of vector and matrix
operations.
The concept of a determinant is already familiar to us from elementary algebra,
where, in solving systems of simultaneous linear equation, we ®nd it convenient to
use determinants. For example, consider the system of two simultaneous linear
equations
)
a 11 x 1 ‡ a 12 x 2 ˆ b 1 ;
…A2:1†
a 21 x 1 ‡ a 22 x 2 ˆ b 2 ;
in two unknowns x 1 ; x 2 where a ij …i; j ˆ 1; 2† are constants. These two equations
represent two lines in the x 1 x 2 plane. To solve the system (A2.1), multiplying the
®rst equation by a 22 , the second by a 12 and then adding, we ®nd
x 1 ˆ b1a 22 b 2 a 12
a 11 a 22 a 21 a 12
:
…A2:2a†
Next, by multiplying the ®rst equation by a 21 , the second by a 11 and adding, we
®nd
x 2 ˆ b2a 11 b 1 a 21
a 11 a 22 a 21 a 12
:
…A2:2b†
We may write the solutions (A2.2) of the system (A2.1) in the determinant form
x 1 ˆ D1
D ;
538
x 2 ˆ D2
D ;
…A2:3†