Mathematical Methods for Physics and Engineering : A Comprehensive Guide

8.9 THE DETERMINANT OF A MATRIX

Suppose the rows of a real 3 × 3 matrixAare interpreted as the components in a given

basis of three (three-component) vectorsa,bandc. Show that one can write the determinant

ofAas

|A|=a·(b×c).

If one writes the rows ofAas the components in a given basis of three vectorsa,bandc,
we have from (8.47) that

|A|=

∣∣

∣

∣∣

∣

a 1 a 2 a 3

b 1 b 2 b 3

c 1 c 2 c 3

∣∣

∣

∣∣

∣

=a 1 (b 2 c 3 −b 3 c 2 )+a 2 (b 3 c 1 −b 1 c 3 )+a 3 (b 1 c 2 −b 2 c 1 ).

From expression (7.34) for the scalar triple product given in subsection 7.6.3, it follows
that we may write the determinant as

|A|=a·(b×c). (8.48)

In other words,|A|is the volume of the parallelepiped defined by the vectorsa,band
c. (One could equally well interpret thecolumnsof the matrixAas the components of
three vectors, and result (8.48) would still hold.) This result provides a more memorable
(and more meaningful) expression than (8.47) for the value of a 3×3 determinant. Indeed,
using this geometrical interpretation, we see immediately that, if the vectorsa 1 ,a 2 ,a 3 are
not linearly independent then the value of the determinant vanishes:|A|=0.

The evaluation of determinants of order greater than 3 follows the same general

method as that presented above, in that it relies on successively reducing the order

of the determinant by writing it as a Laplace expansion. Thus, a determinant

of order 4 is first written as a sum of four determinants of order 3, which

are then evaluated using the above method. For higher-order determinants, one

cannot write down directly a simple geometrical expression for|A|analogous to

that given in (8.48). Nevertheless, it is still true that if the rows or columns of

theN×NmatrixAare interpreted as the components in a given basis ofN

(N-component) vectorsa 1 ,a 2 ,...,aN, then the determinant|A|vanishes if these

vectors are not all linearly independent.

8.9.1 Properties of determinants

A number of properties of determinants follow straightforwardly from the defini-

tion of detA; their use will often reduce the labour of evaluating a determinant.

We present them here without specific proofs, though they all follow readily from

the alternative form for a determinant, given in equation (26.29) on page 942,

and expressed in terms of the Levi–Civita symbolijk(see exercise 26.9).

(i)Determinant of the transpose. The transpose matrixAT(which, we recall,

is obtained by interchanging the rows and columns ofA) has the same

determinant asAitself, i.e.

|AT|=|A|. (8.49)