Calculus: Analytic Geometry and Calculus, with Vectors

2.5 Determinants and applications 87

28 What can be said about the tip of the vector OP if

OP = c,OPI + c2OP2 + c30P3,

where cl, c2, c3 are scalars for which cl + Cl + c3 = 1? Ans.: It lies in the plane

(or each plane) which contains P1, P2, and P3.

2.5 Determinants and applications Rectangular arrays of elements

such as

a21 a22)'

an a12 a23

a21 a22 a23 ,

a31 a32 a33

are called matrices. For the present we may think of the elements a2a as
being numbers. The middle matrix has three rows, the elements of the
second row being a21, a22, a23, and three columns (columns are things that
stand in vertical positions), the elements of the third column being
an, a23, ass. A matrix is square if it contains as many columns as rows,
and in this case the number of rows is called the order of the matrix.
With each square matrix we associate a number which is called the
determinant of the matrix or simply a determinant. The symbols 02, A3,
and A4 appearing in

(2.51) A2 =

ail, axe

a21 'a22

A3=

all

all

a31

a12 a13

all a23

a32 an

A4 =

all all a13 a14

a21 a22 a23 a24

a31 a32 a33 a34

a41 a42 a43 a44

are numbers, not matrices. It *ill, however, be a convenience to say

that an is the element in the third row and second column of the deter-

minant As instead of saying that it is the element in the third row and

second column of the matrix of which As is the determinant. A little time

spent learning about determinants can pay very handsome dividends.

The number A2 is defined by the formula

(2.52) A2 = alla22 - a12a21

This shows how to evaluate determinants of order 2. For example,

1 0

0 1 = 1, a bl =ab-ab=0,

1 2

-3 4

= 4 - (-6) = 10.

The definitions of As and A4 are more complicated, and we introduce
helpful words and notations. To each element a;k of a determinant
there corresponds the minor A;k which remains after the row and column