19.1 Concepts 533
or
Ax 1 = 1 b (19.3)
We saw in Section 17.4 that the equations have a unique solution whendet 1 A 1 ≠ 10 ,
and can then be solved formally in terms of determinants by Cramer’s rule. We also
saw how solutions may be obtained under certain circumstances when Ais singular,
with det 1 A 1 = 10.
When Ais nonsingular (det 1 A 1 ≠ 10 ),A
− 1exists, and premultiplication of the left
side of (19.3) byA
− 1gives
A
− 1(Ax) 1 = 1 (A
− 1A)x 1 = 1 Ix 1 = 1 x
Therefore, premultiplication of both sides of (19.3) byA
− 1gives the unique solution
x 1 = 1 A
− 1b (19.4)
This is equivalent to the use of Cramer’s rule.
EXAMPLE 19.1Solve the equations
2 x 1 − 13 y 1 + 14 z 1 = 18
y 1 − 13 z 1 = 1 − 7
x 1 + 12 y 1 + 12 z 1 = 111
The matrix of the coefficients and its inverse are
Therefore, by equation (19.4),
This is the the result obtained in Example 17.2 by Cramer’s rule.
0 Exercises 1–3
x
y
z
=−
−−
1
21
8145
306
172
−
=
=
8
7
11
1
21
21
42
63
1
22
3
AA=
−
−
,=−
−
−234
013
122
1
21
8145
306
1
1−−
72