Calculus: Analytic Geometry and Calculus, with Vectors

2.5 Determinants and applications 91

has a unique solution, the second system has no solutions, and the third
system has many solutions including the nontrivial one x1 = 2, x2 = -1.
Partly because of these two theorems, determinants are important.
Determinants were originally devised to speed the process of solving
systems of equations whose coefficients are given in decimal form. It is
sometimes said that young algebra students should not be taught to
solve systems of equations by use of determinants because the method
is inefficient and yields too many errors; the method of successive elimi-
nations is much better. This argument is vulnerable, because students

who solve systems of equations by use of determinants acquire facility
in use of determinants. In this course, it is recommended that deter-

minants be used only for purposes for which they are useful.

Problems 2.59

1 Show that

A B

1 2

C

-1 =14A-8B-2C.

(^345)
2 Supposing that Pl(x,,yl) and P2(x2,y 2) are fixed points in E2, show that
the equation
x
xl
X2
Y 1
Yi^1
Y2^1
=0
Ax+By+C=0
and that the graph contains P, and P2. Comment upon the result. Solution:
Expanding (1) in terms of the elements of the first row gives (2). The equation
(1) is satisfied when x = x,, y = y1 and when x = x2, y = y2 because in each
case the determinant has two identical rows. The equation (2) is the equation
of a line unless A = B = 0, that is, unless P, and P2 coincide. If P, and P2 do
coincide, the equation (2) becomes Ox + Oy + 0 = 0 and the graph is the whole
plane.
3 Letting A be the first determinant in the formula
x
X1
X2
xa
y
Yi
Y2
Ys
x-x1 y-y1 z-z1
X1 y, z1
X2 - x1 Y2 - Y1 z2 - z1
xa - x1y8 - y, Z3 - z1
0
1
0
0
show how the formula can be obtained and show that
x-x1 y-yi z-a,
x2-x1 y2 -y1 z2 -z1
x3-x1 ys-y1 Z3-zl