Mathematical Methods for Physics and Engineering : A Comprehensive Guide

8.9 THE DETERMINANT OF A MATRIX

which shows that the trace of a multiple product is invariant under cyclic

permutations of the matrices in the product. Other easily derived properties of

the trace are, for example, TrAT=TrAand TrA†=(TrA)∗.

8.9 The determinant of a matrix

For a given matrixA, the determinant detA(like the trace) is a single number (or

algebraic expression) that depends upon the elements ofA. Also like the trace,

the determinant is defined only forsquarematrices. If, for example,Ais a 3× 3

matrix then its determinant, oforder3, is denoted by

detA=|A|=

∣

∣

∣

∣

∣

∣

A 11 A 12 A 13

A 21 A 22 A 23

A 31 A 32 A 33

∣

∣

∣

∣

∣

∣

. (8.45)

In order to calculate the value of a determinant, we first need to introduce

the notions of theminorand thecofactorof an element of a matrix. (We

shall see that we can use the cofactors to write an order-3 determinant as the

weighted sum of three order-2 determinants, thereby simplifying its evaluation.)

The minorMijof the elementAijof anN×NmatrixAis the determinant of

the (N−1)×(N−1) matrix obtained by removing all the elements of theith

row andjth column ofA; the associated cofactor,Cij, is found by multiplying

the minor by (−1)i+j.

Find the cofactor of the elementA 23 of the matrix

A=





A 11 A 12 A 13

A 21 A 22 A 23

A 31 A 32 A 33



.

Removing all the elements of the second row and third column ofAand forming the
determinant of the remaining terms gives the minor

M 23 =

∣

∣

∣

∣

A 11 A 12

A 31 A 32

∣

∣

∣

∣.

Multiplying the minor by (−1)2+3=(−1)^5 =−1gives

C 23 =−

∣

∣

∣∣A^11 A^12

A 31 A 32

∣

∣

∣∣.

We now define a determinant asthe sum of the products of the elements of any

row or column and their corresponding cofactors,e.g.A 21 C 21 +A 22 C 22 +A 23 C 23 or

A 13 C 13 +A 23 C 23 +A 33 C 33. Such a sum is called aLaplace expansion. For example,

in the first of these expansions, using the elements of the second row of the