Understanding Engineering Mathematics

=

3 ( 1 + 1 )− 2 (− 2 − 1 )

− 8

=

6 + 6

− 8

=−

3

2

Note that in the case when at least one of the valuesbiis non-zero the system of equations
only has a unique solution if the determinant of coefficientsis non-zero. This provides
a simple test for theconsistencyof a system. Ifallbiare zero we say that the system is
homogeneous. In this case, if the determinant of coefficients is non-zero then we simply
get a zero solution for all the variables – the so-called ‘trivial solution’. On the other hand,
if the determinant of coefficients is zero then we get an indeterminate result^00. In this case
it may be possible to get non-zero solutions for some or all of the variables. We consider
homogeneous systems further in Section 13.7.
Geometrically, an equation of the form

ax+by+cz=d

represents a plane in three dimensions (340
➤
). Three equations of this form would
therefore be expected to yield the intersection of three planes and usually this would be
a single unique point corresponding to the unique solution of the system. However, there
are some exceptional cases:

(i) 2 or 3 of the planes parallel, in which case the determinant of coefficients=0and

x,y,zare not uniquely defined.

(ii) 3 planes coincide or have a line in common in which case we again have=0, the
numerators forx,y,z, are all zero, and in this case there are an infinite number of
solutions.

Exercise on 13.5

Solve the system of equations where possible

(i) 3x− 2 y+z= 1 (ii) x+ 2 y− 3 z= 0

x+y+z= 02 x−y+z= 1

2 x−y+ 2 z= 1 x− 3 y+ 4 z= 2

Answer

(i)x=0,y=−^13 ,z=^13 (ii) No solution

13.6 The inverse matrix

Before defining the inverse matrixA−^1 of a square matrixAwe define another matrix,
AdjA, associated withA. The definition seems a bit strange, but all will be revealed
shortly. Theadjointof a square matrixA, denoted byAdjA, is the transpose of the matrix
in which each element ofAis replaced by its corresponding cofactors. Try the problem
quickly to get a proper grasp of this!