Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MATRICES AND VECTOR SPACES

It follows thatanytheorem established for the rows ofAwill apply to the

columns as well, and vice versa.

(ii)Determinant of the complex and Hermitian conjugate. It is clear that the

matrixA∗obtained by taking the complex conjugate of each element ofA

has the determinant|A∗|=|A|∗. Combining this result with (8.49), we find

that

|A†|=|(A∗)T|=|A∗|=|A|∗. (8.50)

(iii)Interchanging two rows or two columns. If two rows (columns) ofAare

interchanged, its determinant changes sign but is unaltered in magnitude.

(iv)Removing factors. If all the elements of a single row (column) ofAhave

a common factor,λ, then this factor may be removed; the value of the

determinant is given by the product of the remaining determinant andλ.

Clearly this implies that if all the elements of any row (column) are zero

then|A|= 0. It also follows that if every element of theN×NmatrixA

is multiplied by a constant factorλthen

|λA|=λN|A|. (8.51)

(v)Identical rows or columns. If any two rows (columns) ofAare identical or

are multiples of one another, then it can be shown that|A|=0.

(vi)Adding a constant multiple of one row (column) to another. The determinant

of a matrix is unchanged in value by adding to the elements of one row

(column) any fixed multiple of the elements of another row (column).

(vii)Determinant of a product.IfAandBare square matrices of the same order

then

|AB|=|A||B|=|BA|. (8.52)

A simple extension of this property gives, for example,

|AB···G|=|A||B|···|G|=|A||G|···|B|=|A···GB|,

which shows that the determinant is invariant under permutation of the

matrices in a multiple product.

There is no explicit procedure for using the above results in the evaluation of

any given determinant, and judging the quickest route to an answer is a matter

of experience. A general guide is to try to reduce all terms but one in a row or

column to zero and hence in effect to obtain a determinant of smaller size. The

steps taken in evaluating the determinant in the example below are certainly not

the fastest, but they have been chosen in order to illustrate the use of most of the

properties listed above.