MATRICES AND VECTOR SPACES
the unique solution. The proof given here appears to fail if any of the solutions
xiis zero, but it can be shown that result (8.130) is valid even in such a case.
Use Cramer’s rule to solve the set of simultaneous equations (8.123).Let us again represent these simultaneous equations by the matrix equationAx=b,i.e.
243
1 − 2 − 2
−33 2
x 1
x 2
x 3
=
4
0
− 7
.
From (8.58), the determinant ofAis given by|A|= 11. Following the discussion given
above, the three Cramer determinants are
∆ 1 =
∣
∣
∣∣
∣
∣
443
0 − 2 − 2
−73 2
∣
∣
∣∣
∣
∣
, ∆ 2 =
∣
∣
∣∣
∣
∣
243
10 − 2
− 3 − 72
∣
∣
∣∣
∣
∣
, ∆ 3 =
∣
∣
∣∣
∣
∣
244
1 − 20
− 33 − 7
∣
∣
∣∣
∣
∣
.
These may be evaluated using the properties of determinants listed in subsection 8.9.1
and we find ∆ 1 = 22, ∆ 2 =−33 and ∆ 3 = 44. From (8.130) the solution to the equations
(8.123) is given by
x 1 =22
11
=2,x 2 =− 33
11
=− 3 ,x 3 =44
11
=4,
which agrees with the solution found in the previous example.
At this point it is useful to consider each of the three equations (8.129) as rep-resenting a plane in three-dimensional Cartesian coordinates. Using result (7.42)
of chapter 7, the sets of components of the vectors normal to the planes are
(A 11 ,A 12 ,A 13 ), (A 21 ,A 22 ,A 23 )and(A 31 ,A 32 ,A 33 ), and using (7.46) the perpendic-
ular distances of the planes from the origin are given by
di=bi
(
A^2 i 1 +A^2 i 2 +A^2 i 3) 1 / 2 fori=1,^2 ,3.Finding the solution(s) to the simultaneous equations above corresponds to finding
the point(s) of intersection of the planes.
If there is a unique solution the planes intersect at only a single point. Thishappens if their normals are linearly independent vectors. Since the rows ofA
represent the directions of these normals, this requirement is equivalent to|A|=0.
Ifb=( 000 )T= 0 then all the planes pass through the origin and, since there
is only a single solution to the equations, the origin is that solution.
Let us now turn to the cases where|A|= 0. The simplest such case is that inwhich all three planes are parallel; this implies that the normals are all parallel
and soAis of rank 1. Two possibilities exist:
(i) the planes are coincident, i.e.d 1 =d 2 =d 3 , in which case there is an
infinity of solutions;
(ii) the planes are not all coincident, i.e.d 1 =d 2 and/ord 1 =d 3 and/or
d 2 =d 3 , in which case there are no solutions.