Calculus: Analytic Geometry and Calculus, with Vectors

90 Vectors and geometry in three dimensions

What this and similar formulas say is that we do not change the value

of a determinant when we add a constant multiple of the elements of one

row (or column) to the elements of another row (or column). For

example, we obtain the first equality in

2 -3^121121 -5

1 -2 3 1 0 3 1 0 0

3 3 -5 3 9 -5 3 9 -14

by adding 2 times the elements of the first column to the elements of the

second column, and then we obtain the second equality by adding -3

times the elements of the first column to the elements of the last column.

As we have seen, this reduces the problem of evaluating a determinant

of order 3 to the problem of evaluating a determinant of order 2.

The following two theorems, and their obvious modifications involving

systems containing two or more than three equations, are very important.

Theorem 2.57 The system of equations

a11x1+ a12x2 + a13x3 = yl

a21x1 + a22x2 + a23x3 = y2

a31x1 + a32x2 + a33x3 = y3

has a unique solution (is satisfied by one and only one set of numbers

x1i X2, x3) if and only if

all a12 a13

a21 a22 a23

a31 a32 a33

0,

that is, the determinant of the coefficients is different from 0.

Theorem 2.58 The system of equations

allxl + a12x2 + a13x3 = 0

a21x1 + a22x2 + a23x3 = 0

a31xl + a32x2 + a33x3 = 0

has a nontrivial solution (a solution for which x1, x2, x3

and only if

all a12 a13

a21 a22 a23

a31 a32 a33

= 0,

are not all 0) if

that is, the determinant of the coefficients is 0.

Proofs of these theorems belong in books andcourses in algebra, but

everybody can observe that the first system of equations

2x1 + 4x2 = 8 2x1 + 4x2 = 8 2x1 + 4x2 = 0

x1 - 2x2 = 1, xl + 2x2 = 0, xl + 2X2 = 0